Instructions: Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

We're about to embark on 3 months of mathematics together. Although this is the start of our mathematical journey together, each of us comes to this class with our own mathematical past. To help seat this semester's work in the larger story of your mathematical journey, I'd like you to write a 300-400 word mathematical autobiography. This autobiography will not be shared with other students (unless you choose to do so). After reading autobiographies I might share some summary information; I might also want to post excerpts from some people's essays, but I'd only do this after receiving permission to do so.

In your autobiography, you might choose to explore some of the following:
- An example of a successful/satisfying mathematical experience in your past, as well as an example of an unsuccessful/disheartening mathematical experience.
- An accounting of the skills which facilitate your mathematical work, as well as a reflection on the obstacles which make working on mathematics more difficult for you.
- What qualities of mathematics do you find fun? (or boring, or scary, or dull, or mystical, etc.)
- How does this class fit into your mathematical journey? your overall academic journey? your overall life journey?
Given the word limitation, it's unlikely that you can pursue all of these prompts. Instead, focus on whatever questions speak the most to you, or chart your own path if you prefer.
[NB. It's probably easiest for you to write your autobiography electronically and then print it out. You can just submit the printed version.]
Stop by Andy's office (or come see him during an office hour) and introduce yourself during the second week of classes. You can do this as a part of a larger group if you like. After you've done this, you can write the sentence ``I met with Andy" as your answer to this problem and receive full credit. If you can't make it during office hours, or for some other reason aren't able to drop by during the second week of classes, email Andy to set up some other time to meet.
In class we discussed that there's some ambiguity in Rotman's definition of subring. In order for $S \subseteq R$ to be a subring, there were two reasonable interpretations for Rotman's statement "$1 \in S$": (A) for $S$ to be a subring of $R$, we need $1_R \in S$ (i.e., $S$ contains the multiplicative identity of $R$), and (B) for $S$ to be a subring of $R$, there needs to be some $1_S \in S$ so that $1_S s = s$ for all $s \in S$.
1. Assuming Rotman's distaste for the zero ring indicates that he'd be unwilling to call the zero ring a subring, and assuming the Exercise 5 in the text is correct, show that the definition Rotman intends must (A).
2. ($\star$) In the section "Homomorphisms and Ideals," Rotman makes an offhand comment about a certain object not being a subring. Explain how this comment proves that the definition Rotman intends must be (A).
Complete Exercise 6 in the text.
Complete Exercise 7 in the text.
Complete Exercise 8 in the text.
Complete Exercise 9 in the text.
Complete Exercise 13 in the text.
Complete Exercise 17 in the text.

Instructions: Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Recall that in the construction of the fraction field of a domain $R$, we introduced the relation on $R \times (R-\{0_R\})$ by $(a,b) \sim (c,d)$ if and only if $ad=bc$.
1. Prove that this gives an equivalence relation.
2. Following the book's convention, let $a/b$ denote the equivalence class of $(a,b)$. Prove that the operations $\oplus$ and $\otimes$ given by $a/b\oplus c/d = (ad+bc)/bd$ and $(a,b)\otimes (c,d) = (ac)/(bd)$ are well-defined. (Note: this includes checking that the outputs of $\oplus$ and $\otimes$ are represented by elements in $R \times (R-\{0_R\})$.)
3. Does this relation provide an equivalence relation for all rings $R$? More explicitly, if $R$ is any ring, and we define the relation on $R \times (R-\{0_R\})$ by $(a,b) \sim (c,d)$ iff $ad=bc$, is his an equivalence relation? If so, prove it; if not, give an explicit counterexample.
Let $R$ be a domain, and let $F$ be a field. Suppose there exists some injection $i:R \to F$. It is a fact (that you don't have to prove) that $i(R)$ is a subring of $F$ which is isomorphic to $R$. Prove that there exists some field $\hat F$ with $i(R) \subseteq \hat F \subseteq F$ so that $\hat F \simeq \text{Frac}(R)$.
[N.B. This is one of the ways of interpretting the statement "$\text{Frac}(R)$ is the smallest field containing $R$."]
The book's definition of greatest common divisor of two polynomials is a bit different from the corresponding definition of greatest common divisor over $\mathbb{Z}$, where the definition really does coincide with the name (i.e., it is the greatest amongst all the common divisors). In the integral case, there is a theorem that says if $d = \gcd(n,m)$ and if $c \mid n$ and $c\mid m$, then it follows that $c \mid d$. One of the questions we considered in class was whether one could modify the definition of greatest common divisor in the book so that it sounded more like the definition we're accustomed to, and whether this modified definition would simply be a way of restating the more clunky definition from the book. Given that the best reasonable metric for "bigness" of a polynomial is degree, the straight-forward modification that fits the bill is this:
If $R$ is a domain and $f(x),g(x) \in R[x]$ are given, then we'll define $\widetilde{gcd}(f(x),g(x))$ to be a polynomial $\tilde d(x)$ that satisfies
1. $\tilde d(x)$ is a common divisor of $f(x)$ and $g(x)$
2. for any common divisor $c(x)$ of $f(x)$ and $g(x)$, we have $\partial(\tilde d(x)) \geq \partial(c(x))$, and
3. $\tilde d(x)$ is monic.
We'll show that these two notions coincide in the special case of polynomials with coefficients from a field $F$ (which, frankly, is the context we'll be most interested in going forward).
1. Suppose that $z(x),y(x),w(x) \in F[x]$ are given so that $z(x) \mid w(x)$ and $y(x) \mid w(x)$. Write $v(x) = \gcd(z(x),y(x))$. Prove that there exists $u(x) \in F[x]$ with $u(x)v(x) = z(x)y(x)$, and so that $u(x) \mid w(x)$. [Hint: Bezout]
2. Let $f(x),g(x) \in F[x]$ be given. Prove that $\widetilde{gcd}(f(x),g(x))$ exists and is unique.
3. Prove that $\gcd(f(x),g(x)) = \widetilde{gcd}(f(x),g(x))$.
Complete Exercise 29.
Complete Exercise 32.
Complete Exercise 38.
Consider the quotient ring $\mathbb{Z}_2[x]/(x^3+x^2+1)$.
1. Write out the multiplication table for this ring. (N.B.: to start this problem, it makes sense to begin by writing out all (distinct) cosets in this quotient. Then you can consider how to multiply them together.)
2. Is $\mathbb{Z}_2[x]/(x^3+x^2+1)$ a field?

Instructions: Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Our book's definition of degree fails to assign a degree to the zero polynomial. Instead, one could define a degree function on all of $R[x]$ by using the "usual" definition of degree for non-zero polynomials, and assigning the zero polynomial some appropriately chosen value. The only consideration in making this assignment is that the value we assign to the degree of the zero polynomial shouldn't cause it to run afoul of results about degree we already know.
1. Let $z \in \mathbb{Z}$ be given, and suppose we assign $\partial(0)=z$. Show that this choice runs counter to the familiar result that the degree of a product of polynomials is the sum of the degrees of those polynomials.
2. Suppose we assign $\partial(0) = -\infty$. Prove that with this extended definition of degree, we still have that the degree of a product of polynomials is the sum of the degrees of those polynomials.
[NB. From now on, feel free to use $\partial(0) = -\infty$. One of the benefits of this choice is that in using the division algorithm, we no longer need to break into two cases: one where $r(x)=0$, and the other when $\partial(r(x))$ is "small."]
Let $F$ be a field, and let $\Phi:F[x] \to F^F$ be the function which assigns each $f(x) \in F[x]$ the associated polynomial function: $\Phi(f(x)) = f$.
1. Is it true that if $\Phi(f(x)) = \Phi(g(x))$, then $F[x]/(f(x)) \simeq F[x]/(g(x))$? If so, prove it; if not, give an explicit (fully justified) counterexample.
2. Let $F = \mathbb{Z}_p$. Prove that $\ker(\Phi) = (x^p-x)$. [Hint: show that if $\partial(g(x)) < p$, then $\Phi(g(x))$ is not the zero function.]
3. Let $F = \mathbb{Z}_p$. Prove that $\Phi$ is surjective. [Hint: count.]
Suppose that $f(x)$ and $g(x)$ are nonzero polynomials in $F[x]$, where $F$ is a field. Prove that $\alpha \text{lcm}(f(x),g(x))\gcd(f(x),g(x)) = f(x)g(x)$, where $\alpha$ is the product of the leaning coefficients of $f(x)$ and $g(x)$. [Hint: Show that "$f(x)g(x)/c \gcd(f(x),g(x))$" satisfies the properties of the lcm. Assignment 2, Problem 3 might be useful.]
Complete Exercise 44 in the text.
Complete Exercise 48 in the text.
Complete Exercise 49 in the text.

Instructions: Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Let $F = \text{Frac}(\mathbb{Z}_5[x])$, and consider the ring $F[t]$. (These are polynomials in the "variable" $t$, whose coefficients are elements of $F$.) Show that $f(t) = t^5-x \in F[t]$ has no roots in $F$, but it has repeated roots in a field $E$ which contains (an isomorphic copy of) $F$. Indeed, show that if $E$ is a field containing $F$ and which has a root $\alpha \in E$ of $f(t)$, then $E$ splits $f(t)$, and $f(t) = (t-\alpha)^5$.
[N.B. You may use the fact that there's a (well-defined) notion of degree for elements of rational function fields. If $\frac{h(x)}{g(x)}$ is a given element, then it's degree is $\partial(h(x))-\partial(g(x))$. This notion of degree also obeys the "degree is additive on products" property we're used to for degree of "regular" polynomials (e.g., Exercise 13(i) in the text). Be aware that for elements in $F[t]$ above, there are two notions of degree in play: one measures "degree in $t$" (applicable to an element of $F[t]$) and the other measures "degree in $x$" (applicable to the coefficients of an element of $F[t]$). So, for example, a polynomial like $\frac{x^2+2}{x^3+x+1}t^2-\frac{4}{x^2}t+\frac{x^4+x+1}{x^2+x} \in F[t]$ has degree $2$ in $t$, and its coefficients (for $t^2$, $t$ and $1$) have degree $-1, -2$ and $2$ in $x$ (respectively).]
Suppose that $F$ is a field of characteristic $0$, and that $p(x)$ is irreducible. Prove that $p(x)$ has no repeated roots (in any field $E$ containing $F$).
1. Prove that $p^m-1 \mid p^n -1$ if and only if $m \mid n$.
2. Prove that if $m \nmid n$, then there is no field with $p^n$ elements which contains a subfield with $p^m$ elements. [Hint: If $\mathbb{F}$ is a field, then $\mathbb{F}\setminus \{0\}$ is a group under multiplication.]
What is the smallest ~~splitting field of~~ field in which $x^4-x \in \mathbb{Z}_3[x]$ splits?
Complete Exercise 63 in the text. [Note: the intention is that you prove the second sentence, then use that sentence to prove the third sentence.]

Instructions: Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Suppose that $F$ is a subfield of $E$. Prove that the prime field of $F$ is the same as the prime field of $E$.
[N.B.: The upshot of this problem is that characteristic is unchanged under field extension.]
Let $p$ be prime and $n \in \mathbb{N}$ be given. Recall that we constructed a field $\mathbb{F}_{p^n}$ with $p^n$ elements as follows: for a field $E$ containing $\mathbb{Z}_p$ over which $x^{p^n}-x \in \mathbb{Z}_p[x]$ splits, the field $\mathbb{F}_{p^n}$ is the set of elements of $E$ which satisfy $\alpha^{p^n} = \alpha$. Prove that if $m \mid n$, then $\mathbb{F}_{p^n}$ contains a field with $p^m$ many elements.
[Hint: show that $\mathbb{F}_{p^n}$ contains precisely $p^m$ elements which are roots to $f(x) = x^{p^m}-x$, and that these roots form a field. Last week's pset might be a useful tool. Though you don't have to prove it, observe that this really means that $\mathbb{F}_{p^n}$ contains $\mathbb{F}_{p^m}$ itself.]
Recall that penguin's (more explicit) construction of $\mathbb{F}_{p^n}$ followed the following template: take the polynomial $x^{p^m}-x \in \mathbb{Z}_p[x]$ and find an irreducible (non-linear) factor $p(x)$ of $x^{p^m}-x$; then consider the field $K = \mathbb{Z}_p[x]/(p(x))$. While $K$ contains at least one root of $p(x)$, it's not clear a priori that $K$ will split the entire polynomial $p(x)$, let alone $x^{p^m}-x$. As it happens, it will be the case that $K$ splits $x^{p^m}-x$, but we don't (yet) have quite enough technology to show this. Instead, in this problem we'll verify this fact holds in one special case.

Suppose $p=2$ and $m = 3$, and choose your favorite non-linear irreducible factor $p(x)$ of $x^8-x \in \mathbb{Z}_2[x]$. Show that the field $E = \mathbb{Z}_2[x]/(p(x))$ splits $x^8-x$.
[Note: this problem will require you to factor $x^8-1$ as much as possible over $\mathbb{Z}_2[x]$. Given that any irreducible polynomial $p(x)\in\mathbb{Z}_2[x]$ has $|\mathbb{Z}_2[x]/(p(x))| = 2^{\partial(p(x))}$, you should be able to make a guess about the degrees of non-linear factors of $x^8-x$. The results of quiz 4 might then be useful for "guessing" what the factorization could be.]
Throughout this problem, we'll write $F$ for a field with $p^n$ elements (where of course $p$ is prime and $n \in \mathbb{N}$).
1. Show that the function $\phi:F \to F$ given by $\phi(a) = a^p$ is a field isomorphism.
2. Suppose that $f(x) = \sum c_i x^i \in F[x]$ is irreducible, and that there exists some $i$ with $p \nmid i$ and $c_i \neq 0$. Prove that $f(x)$ has no repeated roots (in any field $E$ containing $F$).
3. Suppose that $f(x) = \sum c_i x^i \in F[x]$ has $\partial(f(x))>0$, and that for all $i$ with $c_i \neq 0$ we have $p \mid i$. Show that $f(x)$ is reducible. [Hint: write $f(x) = g(x^p)$ for some $g(x) \in F[x]$, then use the first part of this problem to write the coefficients of $g$ as $p$th powers; finally, apply Lemma 32 to produce a nontrivial factorization.]
($\star$) Complete problem 67 in the text. [Note: To use the hint the book gives, you need to justify why any factorization of $f(x)$ into quadratic factors would have the property that the linear coefficients of those polynomials are additive inverses of each other.]

Instructions: Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Recall in class we used the notation $F[\alpha]$ to denote the set of "polynomials in $\alpha$," whereas $F(\alpha)$ is the "field generated by $\alpha$ over $F$." More explicitly, this means $$F[\alpha] = \{f_0+f_1\alpha+\cdots+f_n \alpha^n: n \in \mathbb{N} \text{ and } f_0,\cdots,f_n \in F\},$$ and that $F(\alpha) = \text{Frac}(F[\alpha])$. We also saw that if $\alpha$ is algebraic, then $F(\alpha) = F[\alpha]$.
Prove that if $F(\alpha) = F[\alpha]$, then $\alpha$ is algebraic.
Complete problem 74 from the text.
($\star$) Suppose that $\alpha,\beta \in E$ are algebraic over $F$, where $E/F$ is a field extension.
1. Prove $$[F(\alpha,\beta):F] \leq [F(\alpha):F][F(\beta):F].$$ [Hint: Prove that $[F(\beta):F] \geq [F(\alpha,\beta):F(\alpha)]$.]
2. Give an (explicit) example of a field $F$ and algebraic elements $\alpha$ and $\beta$ so that the inequality above is strict.
[N.B.: Of course one can apply this result inductively to show that if $\alpha_1,\cdots,\alpha_n \in E$ are algebraic, then $[F(\alpha_1,\cdots,\alpha_n):F] \leq \prod_{i=1}^n [F(\alpha_i):F].$ Moving forward, you can feel free to use this stronger result without proof.]
Write $\mathbb{A}$ for the set of $\alpha \in \mathbb{C}$ which are algebraic over $\mathbb{Q}$.
1. Prove that $\mathbb{A}$ is a field. [Hint: the previous argument tells you that if $\alpha$ and $\beta$ are algebraic, then there is a finite extension of $\mathbb{Q}$ that contains both.]
2. Prove that $[\mathbb{A}:\mathbb{Q}] = \infty$. [Hint: argue that for any $N \in \mathbb{N}$ there exists some $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$ so that $[K:\mathbb{Q}] = N$.]
The polynomial $x^3-2 \in \mathbb{Q}[x]$ has one real root $\alpha_1 = \root{3}\of{2} \approx 1.2599$, as well as two (non-real) complex roots $\alpha_2 = \root{3}\of{2}\left(\frac{-1 + i\sqrt{3}}{2}\right)$ and $\alpha_3 = \root{3}\of{2}\left(\frac{-1 - i\sqrt{3}}{2}\right)$.
1. Prove that $\mathbb{Q}(\alpha_1)$ does not split $x^3-2$.
2. Prove that $\mathbb{Q}(\alpha_1,\alpha_2)$ does split $x^3-2$.
3. Conclude that the splitting field $K = \mathbb{Q}(\alpha_1,\alpha_2,\alpha_3)$ of $x^3-2$ has $[K:\mathbb{Q}] = 6$.
4. Prove that $\mathbb{Q}(\alpha_2)$ does not split $x^3-2$.

Here are the Solutions.

Relavent content: Videos 1-4.

Instructions:

To earn credit in the class, provide high-quality solutions to any 4 of the problems below. You do not need to complete all problems on this assignment unless you are aspiring for "credit with distinction." (Aside from credit with distinction, you might choose to do more problems simply because working on math problems is amazing, and because Galois theory in particular is the best of all math).

It will be extremely useful to me if you send your solutions along as a PDF file whose filename is of the form "Your_Name_Assignment_Name.pdf". (So if I were submitting my solutions to pset 6, I might submit a file called "Andy_Schutz_Pset_6.pdf")

Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Suppose that $\alpha$ is an algebraic element over $F$.
1. Prove that $\partial(\text{irr}_F(\alpha)) = \partial(\text{irr}_F(\alpha^{-1}))$.
2. Compute $\text{irr}_\mathbb{Q}\left(\sqrt{2}+\sqrt{5}\right)$ and $\text{irr}_\mathbb{Q}\left((\sqrt{2}+\sqrt{5})^{-1}\right)$.
Suppose that $E/F$ is a finite extension of the form $E = F(\alpha_1,\alpha_2,\cdots,\alpha_n)$. Prove that an $F$-basis for $E$ is given by $$\{\alpha_1^{e_1}\cdots\alpha_n^{e_n}: 0 \leq e_i < \partial(\text{irr}_{F(\alpha_1,\cdots,\alpha_{i-1})}(\alpha_i)\}$$
Let $\alpha_1 = \sqrt[3]{2}$, $\alpha_2 = \omega \sqrt[3]{2}$, and $\alpha_3 = \omega^2 \sqrt[3]{2}$, where $\omega = \frac{-1+\sqrt{-3}}{2}$. We've seen already that $E = \mathbb{Q}(\alpha_1,\alpha_2)$ is the splitting field for $x^3-2$, and by the problem above we know that $\{1,\alpha_1,\alpha_1^2,\alpha_2,\alpha_1\alpha_2,\alpha_1^2\alpha_2\}$ is a $\mathbb{Q}$-basis for $E$. For the element $\sigma \in \text{Gal}(E/\mathbb{Q})$ defined by $\sigma(\alpha_1) = \alpha_2$ and $\sigma(\alpha_2) = \alpha_3$, express the value of $$\sigma(q_1 + q_2 \alpha_1 + q_3 \alpha_1^2 + q_4\alpha_2 + q_5\alpha_1\alpha_2 + q_6\alpha_1^2\alpha_2)$$ in terms of the $\mathbb{Q}$-basis $\{1,\alpha_1,\alpha_1^2,\alpha_2,\alpha_1\alpha_2,\alpha_1^2\alpha_2\}$. (Of course, $q_1,q_2,q_3,q_4,q_5,q_6 \in \mathbb{Q}.)

(If you're feeling up for it, you can then use this to write a matrix representation for $\sigma$.)
Let $\alpha_1 = \sqrt[3]{2}$, $\alpha_2 = \omega \sqrt[3]{2}$, and $\alpha_3 = \omega^2 \sqrt[3]{2}$, where $\omega = \frac{-1+\sqrt{-3}}{2}$. We've seen already that $E = \mathbb{Q}(\alpha_1,\alpha_2)$ is the splitting field for $x^3-2$. Let $L=\mathbb{Q}(\omega)$.
1. Prove that $\mathbb{Q} \subseteq L \subseteq E$.
2. Prove that $L$ is the splitting field of a polynomial $f(x) \in \mathbb{Q}[x]$.
3. Prove that $E$ is a simple extension of $L$.
4. Give a $\mathbb{Q}$-basis for $E$ inspired by ``climbing to $E$ via $L$." (You can use the second problem of this pset even if you don't complete it.)
5. For the element $\sigma \in \text{Gal}(E/F)$ defined by $\sigma(\alpha_1) = \alpha_2$ and $\sigma(\alpha_2) = \alpha_3$, determine how $\sigma$ acts on the basis you wrote down in the previous part.
6. Determine how the element $\sigma \in \text{Gal}(E/F)$ above acts on a general element of $E$ expressed relative to your $\mathbb{Q}$-basis from part (d).
Let $K/F$ be a field extension, and let $f \in F$.
1. Show that a splitting field for $x^n-f$ must contain a splitting field for $x^n-1$.
2. Suppose that $K$ splits $x^n-1$. Show that if $K$ contains one root of $x^n-f$, then $x^n-f$ splits completely in $K$.
Complete problem 80 in the text.
Suppose that $\theta \in \mathbb{R}$. Use induction and angle sum formulas to prove De Moivre's theorem: $$(\cos(\theta)+i\sin(\theta))^k = \cos(k\theta)+i\sin(k\theta)$$ (where here $i = \sqrt{-1}$). Use this to conclude that $\{\cos\left(\frac{2\pi k}{n}\right)+i\sin\left(\frac{2\pi k}{n}\right): 0 \leq k < n\}$ are $n$ distinct solutions to $x^n-1$.

Here are the Solutions.

Relavent content: Videos 5-6.

Instructions:

To earn credit in the class, provide high-quality solutions to any 4 of the problems below. You do not need to complete all problems on this assignment unless you are aspiring for "credit with distinction." (Aside from credit with distinction, you might choose to do more problems simply because working on math problems is amazing, and because Galois theory in particular is the best of all math).

It will be extremely useful to me if you send your solutions along as a PDF file whose filename is of the form "Your_Name_Assignment_Name.pdf". (So if I were submitting my solutions to pset 6, I might submit a file called "Andy_Schutz_Pset_6.pdf")

Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Let $F = \mathbb{Z}_5(x)$. We have already seen that $f(t) = t^5-x \in F[t]$ is irreducible. Let $K/F$ be the splitting field for $f(t)$. Prove that $\text{Gal}(K/F) = \{\text{id}_K\}$.
Complete problem 78, with the additional subquestion (iii) Given an example where $n \nmid |\text{Gal}(E/F)|$.
Complete problem 79. (Omit the part of (ii) that has to deal with regular polygons.)
Complete problem 81.
Throughout this problem, we write $\omega_n$ for a primitive $n$th root of unity. The inspiration for this problem is the theorem we called "Galois groups of root-of-unity extensions," which provides an injection $\text{Gal}(F(\omega_n)/F) \hookrightarrow U(\mathbb{Z}_n)$.
1. It is a fact (that you don't have to prove) that $\partial(\text{irr}_\mathbb{Q}(\omega_n)) = |U(\mathbb{Z}_n)|$. Use this to prove that $\text{Gal}(\mathbb{Q}(\omega_n)/\mathbb{Q}) \simeq U(\mathbb{Z}_n)$.
2. Prove that $\text{Gal}(\mathbb{Z}_2(\omega_7)/\mathbb{Z}_2)$ is (isomorphic to) a proper subgroup of $U(\mathbb{Z}_7)$. [Hint: $x^8-x = x(x^7-1)$.]
[N.B. In both cases, it might be useful to argue that adjoining the given $\omega_n$ to the given field produces a splitting field for a seperable polynomial.]
In Video 6 we said that if $E$ is the splitting field for $x^p-c \in F[x]$, where $F$ is a field that contains a primitive $p$th root of unity $\omega_p$, then $$\text{Gal}(E/F)=\begin{cases}\{\text{id}_E\},&\text{ if }c\in F^{\# p}\\\mathbb{Z}_p,&\text{ if }c \not\in F^{\# p}.\end{cases}$$ In this problem we work to verify this assertion.
1. Prove that $c \in F^{\# p}$ implies $\text{Gal}(E/F) = \{\text{id}_E\}$ by showing that $E = F$.
2. Prove that $c \not\in F^{\# p}$ implies $\text{Gal}(E/F)$ by showing the contrapositive. [Hint: Based on other results in video 6, if $\text{Gal}(E/F) \neq \mathbb{Z}_p$, then what must $\text{Gal}(E/F)$ be? Use what you know about roots of unity in $F$ to argue that $x^p-c$ is separable, then use this to connect the size of $\text{Gal}(E/F)$ to the degree $[E:F]$.]
The motivation for this question is the following. The result we called Kummer Theory (Part I) gives us a way to produce extensions of a field $F$ with Galois group $\mathbb{Z}_p$, at least assuming $F$ contains a $p$th root of unity. However, if $\text{char}(F) =p$, then primitive $p$th roots of unity can't exist (even in an extension of $F$). So how can we create extensions of a field $F$ of characteristic $p$ whose Galois group is $\mathbb{Z}_p$? The answer is what is often called Artin-Schreier Theory; you'll prove ``part 1" of Artin-Schreier theory in the problem below.
Suppose that $\text{char}(F) = p$. For $c \in F$, define $f(x) = x^p-x-c \in F[x]$. Prove that if $E$ is the splitting field for $f(x)$, then $\text{Gal}(E/F) \hookrightarrow \mathbb{Z}_p$. [Hint: argue that if $\alpha$ is one root of $f$, then the full set of roots is $\{\alpha+k\}$, where $k$ ranges over the elements in the prime field of $F$. Then argue that $f$ is irreducible; for this, it will help to consider the coefficient of the second-highest degree term.]
[In fact, one can prove that $\text{Gal}(E/F) \simeq \mathbb{Z}_p$ if and only if $c \not\in \{l^p-l: l \in F\}$, though you don't need to do that for this problem.]

Here are the Solutions.

Relavent content: Videos 7,8,10

Instructions:

To earn credit in the class, provide high-quality solutions to any 4 of the problems below. You do not need to complete all problems on this assignment unless you are aspiring for "credit with distinction." (Aside from credit with distinction, you might choose to do more problems simply because working on math problems is amazing, and because Galois theory in particular is the best of all math).

It will be extremely useful to me if you send your solutions along as a PDF file whose filename is of the form "Your_Name_Assignment_Name.pdf". (So if I were submitting my solutions to pset 6, I might submit a file called "Andy_Schutz_Pset_6.pdf")

Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Complete the proof of "Equivalent characterizations for Galois extensions" by proving the following: if $E = F(\alpha_1,\cdots,\alpha_n)$ for algebraic $\alpha_i \in E$, and if $E/F$ has the property that all irreducible $p(x) \in F[x]$ with some root in $E$ satisfy the condition that $p(x)$ splits in $E$, then $E$ is the splitting field for $\prod_{i=1}^n \text{irr}_F(\alpha_i)$.
Complete problem 83 in the text. [This problem is in the "Solvable by radicals" section, but you don't need the technology from that section to answer this question.]
Complete problem 84 in the text. [This problem is in the "Solvable by radicals" section, but you don't need the technology from that section to answer this question.] Note: it appears that what the author intends when he says "as in Exercise 83" is that the field $K$ is the smallest splitting field that contains $B$.
Complete problem 86 in the text by showing that $E/B$ satisfies conditions (2) and (3) in "Equivalent conditions for Galois extensions." [Note: obviously we've proven that if $E/B$ satisfies condition (2), then it must also satisfy condition (3). I want proofs of each condition that don't explicitly rely on the statement of "Equivalent conditions for Galois extensions."]
The following problems all relate to extensions of degree $2$.
1. Complete problem 87 in the text.
2. Prove that if $\text{char}(F) \neq 2$ and $[E:F] = 2$, then $E$ must be the splitting field a polynomial of the form $x^2-f \in F[x]$.
3. Give an example of an extension $E/F$ with $\text{char}(F)=2$ and $[E:F] = 2$, and yet $E/F$ is not Galois.
4. Give an example of an extension $E/F$ with $\text{char}(F)=2$ and $[E:F]=2$, and yet $E/F$ is Galois. Explain why $E$ is not the splitting field of a polynomial of the form $x^2-f \in F[x]$.
Complete problem 88 in the text.
Complete problem 90 in the text.

Here are the Solutions.

Relavent content: Videos 7,8,10

Instructions:

To earn credit in the class, provide high-quality solutions to any 4 of the problems below. You do not need to complete all problems on this assignment unless you are aspiring for "credit with distinction." (Aside from credit with distinction, you might choose to do more problems simply because working on math problems is amazing, and because Galois theory in particular is the best of all math).

It will be extremely useful to me if you send your solutions along as a PDF file whose filename is of the form "Your_Name_Assignment_Name.pdf". (So if I were submitting my solutions to pset 6, I might submit a file called "Andy_Schutz_Pset_6.pdf")

Your answers should always be written in prose form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. Of course your work should be legible and neat.

When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Let $\Phi_n(x) = \prod_{k \in U(\mathbb{Z}_n)} (x-\omega_n^k)$, where $\omega_n = \cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$.
1. Show that for prime values of $n$, this construction for $\Phi_p(x)$ agrees with the construction we gave earlier in the semester.
2. Show that $\Phi_n(x) \in \mathbb{Q}[x]$. [Hint: Let $E$ be the splitting field, and $G$ its Galois group. Prove that for any $\sigma \in G$ we have $\sigma^*(\Phi_n(x)) = \Phi_n(x)$.]
  [N.B.: In fact, it turns out that $\Phi_n(x)$ is irreducible, so that $\Phi_n(x) = \text{irr}_\mathbb{Q}(\omega_n)$. Compare to Pset 8, problem 5.]
3. Show that for odd $n$ we have $\Phi_{2n}(x) = \Phi_n(-x)$.
Suppose that $E/F$ is Galois and $[E:F] = 8$. Show that the number of intermediate fields $F \subseteq K \subseteq E$ with $[K:F]=4$ is equal to either $1$, $3$, $5$ or $7$. [Hint: there are, up to isomorphism, $5$ groups of order $8$.]
Let $f(x) = (x^2-2)(x^2-3)(x^2-5) \in \mathbb{Q}[x]$, and let $E$ be its splitting field. Compute $\text{Gal}(E/\mathbb{Q})$, and draw the lattices for the subgroups of the Galois group and the intermediate extensions in $E/\mathbb{Q}$.
Let $f(x) = x^8-3 \in \mathbb{Q}[x]$, and let $E$ be its splitting field. Compute $\text{Gal}(E/\mathbb{Q})$, and draw the lattices for the subgroups of the Galois group and the intermediate extensions in $E/\mathbb{Q}$.
Let $F$ be a field of characteristic different from $2$.
1. If $E = F(\sqrt{D_1},\sqrt{D_2})$ where $D_1,D_2 \in F$ have the property that none of $D_1,D_2$ or $D_1D_2$ are a square in $F$, prove that $E/F$ is a Galois extension with $\text{Gal}(E/F) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2$.
2. Conversely, suppose that $E/F$ is a Galois extension whose Galois group is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. Prove that $E = F(\sqrt{D_1},\sqrt{D_2})$ for $D_1,D_2 \in F$ with none of $D_1,D_2$, nor $D_1D2$ a square in $F$.
Suppose that $f(x) \in \mathbb{Q}[x]$ is a cubic polynomial whose splitting field has a Galois group isomorphic to $\mathbb{Z}_3$. Prove that the roots of $f$ are all real numbers.
Let $E/F$ be a Galois extension, and let $K_1$ and $K_2$ be intermediate fields in this extension. Prove that $K_2$ is conjugate to $K_1$ if and only if $\text{Gal}(E/K_2)$ is conjugate to $\text{Gal}(E/K_1)$.
(Recall that if $H \leq G$, then for each $x \in G$ the set $xHx^{-1} = \{xhx^{-1}: h \in H\}$ is a subgroup, and these subgroups are the conjugates of $H$.)

Here are the Solutions.

Homework Assignments