Presentation Plans
You will cover a wide variety of materials during lecture and discussion sections; your constant attendance is important not only for your understanding, but the understanding of others. If you want a sense for how your presentations are evaluated, please reference the presentation guidelines.
To help frame our progression through the course, below I've listed rough outlines for the lessons which constitute our course. On average, lessons consist of two sections from the text; I expect that most lessons can be presented in 35 minutes or so. As a guide for what could possibly be covered in a given class, our policy will be that we will not complete more than 2 "new" presentations on a given day. This means on a given day a presenter from last class can finish her lecture, and at most two new students will be giving presentations.
Unit 1: Fundamentals of Ring theory
The content of this section is simply a review of the basic definitions concerning rings and polynomials. Some of this material will be familiar to students from Math 305, while some of it will be new.
-
Presenter: everyone Text section: Consult Galian (or your favorite introductory algebra text) Content: The very quick overview of the content of Math 305 is given via 3-minute mini-presentations. The outline for the day is given here. Comments: Each presenter should think carefully about what key features of their assigned topic they want to highlight, particularly since 3 minutes is an extremely short amount of time to talk about anything! It is unreasonable to think we could give a complete review of all topics covered in 305 during the course of a 50 minute course; instead, set your sights on bringing to the foreground the most important part of your assigned topic, and perhaps giving a single illustrative example. It will be useful to reach out to other presenters who you think might have some overlap with your assigned content to avoid unnecessary repetition. -
Presenter: Xinhui Text section: Rings, from the start of the section at the bottom of page 7 until the top three lines of page 10 Content: The basic definitions for rings that we'll use, plus some stock examples. Critically, the notion of a polynomial ring is defined. Comments: The book's definition of a polynomial is different from the standard definition we all think of, and in particular the monomials $x^i$ are defined in a very particular way (inductively in terms of products of the polynomial $(0,1,0,0,\cdots)$ with itself). It might be useful to have an example that illustrates that polynomials from $R[x]$ are distinct from the functions $R \to R$ they induce; that is, it might be useful to have an example or a ring $R$ and two distinct polynomials $f(x),g(x) \in R[x]$ for which $f(r) = g(r)$ for all $r \in R$. -
Presenter: Angela Text section: Rings, from the start of fourth line of page 10 until the end of the section. However, you should omit the paragraphs on page 10 that start "Recall from linear algebra..." and "The problem of finding...". Content: Some terminology associated with polynomials is introduced, and some immediate corollaries to the axioms of a ring are stated. Comments: The book's definition of a polynomial is different from the standard definition we all think of, and in particular the monomials $x^i$ are defined in a very particular way (inductively in terms of products of the polynomial $(0,1,0,0,\cdots)$ with itself). It might be useful to have an example that illustrates that polynomials from $R[x]$ are distinct from the functions $R \to R$ they induce; that is, it might be useful to have an example or a ring $R$ and two distinct polynomials $f(x),g(x) \in R[x]$ for which $f(r) = g(r)$ for all $r \in R$. -
Presenter: Erin Text section: Domains and Fields, from the start of the section through the statement and proof of Theorem 8 Content: Domains and fields --- and their connection to cancellation and multiplicative inverses --- are defined, and some basic examples are given. Comments: -
Presenter: Audrey Text section: Domains and Fields, from the paragraph following the proof of Theorem 8 (which starts "Every field is a domain...") until the end of the section Content: We know that not all elements $r$ of a ring $R$ have a multiplicative inverse in $R$, but under what circumstances does $r$ have a multiplicative inverse in some larger ring $S$? Fraction fields are constructed to provide a larger algebraic object in which multiplicative inverses exist, even if they aren't in the original ring $R$. Comments: The set of rational functions over a field comes up a great deal in the sequel, so this is an important concept to drive home. -
Presenter: Rebecca Text section: Homomorphisms and Ideals (in its entirety) Content: The basic definitions and properties of ring homomorphisms are presented, and ideals (together with their connection to ring homomorphisms) are introduced. Comments: The definition of a homomorphism in the text differs from the definition that many people will have seen for ring homomorphism, in that it insists that a homomorphism must carry a multiplicative identity to a multiplicative identity -
Presenter: Gabby Text section: Quotient Rings (in its entirety) Content: Quotient rings are constructed, and a few examples are given. Comments: Examples 9 and 10 are fairly deep, and warrant a more explicit designation (and presentation) than mere examples. -
Presenter: Khonzoda Text section: Polynomial Rings over Fields, from the start through line 5 of page 26 Content: The ring of polynomials over a field is shown to be a principal ideal domain. Notions of divisibility and gcd/lcm in the context of polynomial rings are introduced, and it is shown that any two polynomials over a ring of the form $F[x]$ has a gcd, and that this gcd can be expressed as a linear combination of the given polynomials. Comments: Notions of divisibility and gcd/lcm in $\mathbb{Z}$ will be familiar to most students. It will be useful to show similarities and differences between this familiar notion and the newly defined notions in the context of polynomial rings. -
Presenter: Rachel Text section: Polynomial Rings over Fields, from Corollary 15 through line 2 of page 28 Content: Further number-theoretic analogs are carried over from $\mathbb{Z}$ over to $F[x]$, namely Euclid's lemma and the Euclidean algorithm. The notion of gcd is also shown to be stable under field extensions. Comments: Several of these results have a striking resemblance to some basic results in number theory, and it might be useful to use student's prior experience with the analogous theorems over $\mathbb{Z}$ as a source of inspiration. -
Presenter: Amy Text section: Polynomial Rings over Fields, from the definition of lcm until the end of the section Content: The last part of this section is a bit of a hodge podge. The notion of least common multiple of two polynomials is given, and an ideal-theoretic interpretation for this concept is given. A relationship between roots of a polynomial and its factorization are then given, which ultimately gives a method for bounding the number of roots of a polynomial in terms of its degree. Finally, the section finishes with a discussion of the function-theoretic interpretation of polynomials. Comments: Since the results in this section are somewhat disparate, you'll want to ensure that you're helping everyone understand how these topics fit into some larger narrative. -
Presenter: Miranda Text section: Prime Ideals and Maximal Ideals, from the beginning of the section through Theorem 28 Content: The notions of prime and maximal ideals are introduced, and their relation to properties of the corresponding quotients are recorded. Comments: This section seems to be particularly thin on examples. -
Presenter: Hannah Text section: Prime Ideals and Maximal Ideals, from Corollary 29 through Theorem 31 Content: For a given field $F$ and irreducible polynomial $p(x)$, the book gives a method for finding a field $K$ which contains "a copy of" $F$ and a root of $p(x)$. The notion of splitting field is introduced, and their existence is verified. A method for testing for "repeated roots" of a polynomial $p(x) \in F[x]$ is given which can be verified within $F[x]$ itself (even if those roots don't exist in $F$!). Comments: The observation at the top of page 35 should be promoted from off-hand comment to a bona fide lemma (or theorem or whatever) with an explicit proof. -
Presenter: Yiran Text section: Prime Ideals and Maximal Ideals, from definition of characteristic until the end of the section Content: The notion of characteristic is introduced, and certain arithmetic properties are proven based on characteristic. Fields with $p^n$ many elements are constructed. Comments: One must be careful when discussing the prime subfield, since it is not necessarily equal to $\mathbb{Z}_p$, but simply isomorphic. The proof of Theorem 33 is quite abstract; it would be nice to see a finite field built more explicitly (a task which requires one to go back and carefully consider how splitting fields are constructed. A field with $4$ elements will be the most tractable to write down, though one could be ambitious and try to reach for a field of order $8$ or $9$. The paragraph before Theorem 33 is an extremely important comment, and deserves to be promoted into something more official looking. -
Presenter: Mitsuki Text section: Irreducible Polynomials, from the start of the section through Lemma 36 Content: A method is given which can be used to show that certain polynomials with coefficients from a domain cannot be reduced. The book then begins to lay the groundwork for answering whether "irreducible over $\mathbb{Z}" indeed implies ''irreducible over $\mathbb{Q}$." Comments: The big theme in this section is trying to come up with ways for checking whether a particular polynomial (in the polynomial ring of a field) is reducible. This question has a nice answer when the degree of the polynomial is $2$ or $3$ (see Exercise 49), so you should fold this into your presentation. And while theorem 34 gives an answer that is fairly broad (i.e., applicable in lots of rings), subsequent results focus in on $\mathbb{Z}$ and $\mathbb{Q}$. The motivation for this is at least partly historical (historically, "solving a polynomial" was equivalent to "solving an integer coefficient polynomial"), but it's at least worth mentioning that the focus is going to be specialized to this classical setting for the majority of this section. -
Presenter: Yining Text section: Irreducible Polynomials, from Corollary 37 until the end of the section Content: Gauss' and Eisenstein's irreducibility criteria are introduced. Comments: Do we need two proofs of Eisenstein's criterion?
Unit 2: Galois theory
In this unit we work towards the main theorem of Galois theory. We will study field extensions and their properties.
-
Presenter: XinhuiKhonzodaText section: Splitting Fields, from the start of the section through Theorem 46 Content: The basic vocabulary around field extensions is established, and some basic properties concerning finite extensions are proved. Comments: It might have been a while since people thought about vector spaces, so the definition at the end of page 50 likely needs to be appropriately supplemented. -
Presenter: Angela Text section: Splitting fields, from Theorem 47 through the discussion preceding Lemma 49 Content: For algebraic elements, an explicit construction for creating a field which contains a root of an irreducible polynomial is provided. Splitting fields are defined and shown to exist. Comments: -
Presenter: AudreyGabbyText section: Splitting Fields, from Lemma 49 through Lemma 50 Content: An important rule concerning extension degrees is developed, and a lemma is established which allows us to extend known field isomorphisms to extensions. Comments: -
Presenter: Rebecca Text section: Splitting Fields, the definition of separability (for a polynomial) through the end of the section Content: The notion of "separability" is discussed in a few contexts, and the total number of extensions of a field isomorphism are determined (under appropriate conditions regarding separability). Comments: The remark preceding Theorem 51 is not necessary for the purposes of our course (the fields of finite characteristic we'll be interested in are all finite), so it can be omitted from our discussion. -
Presenter: Rachel Text section: The Galois group, from the definition of automorphism through example 24 Content: Automorphisms of field extensions are introduced, which gives rise to the Galois group of a field extension. The size of the Galois group of the splitting field of a separable polynomial is computed. Comments: The section starts with a connection between certain geometric symmetries and the Galois group. This isn't relevant to us, so you should skip it. -
Presenter: Amy Text section: The Galois group, from Lemma 57 to the end of the section Content: If we have an intermediate field $K$ within $E/F$, how do the Galois groups $\text{Gal}(E/F)$, $\text{Gal}(E/K)$ and $\text{Gal}(K/F)$ relate to each other? This section answers that question. Comments: -
Presenter: Xin Text section: Roots of Unity, from the start of the section through the definition of Lemma 66 and the one-sentence remark that follows it. Please skip the discussion about "square" elements (including Corollary 65) and Example 26. You will be required to also define the notion of "primitive root of unity" (which appears on page 68) in order to follow some of the proofs the book gives. Content: With the help of some basic facts about cyclic groups, this section works towards arguing that the nonzero elements in a finite field form a cyclic group, and considers what this means when it comes to generating finite fields as "primitive" extensions of $\mathbb{Z}_p$. Comments: Lemma 59 is the fundamental theorem of cyclic groups from 305; you should feel free to state the result but skip the proof. The book isn't as explicit in drawing some connections as it might be. For instance, the section is about roots of unity, but it never explicitly defines what this means. Part of the challenge for you in this section is to make certain "hidden" parts of this section explicit. As another example, the proof of Corollary 64 leaves a lot for the reader to fill in, in particular why $(q-1)$st roots of unity are relevant in the analysis of $GF(q)$. Indeed, why are finite fields being discussed in a section that's supposedly focused on fields we get by adjoining roots of unity to some ground field? (It's even true that the statement of Corollary 64 isn't as explicit as it could be, since the proof is fairly explicit in telling us the $\alpha$ for which $GF(p^n) = \mathbb{Z}_p(\alpha)$; again, this is a deficiency you should aim to remedy.)
As in all sections, think carefully about the story you are telling in your section, and how it relates to things we've done and sections that follow yours.
-
Presenter: Hannah Text section: Roots of Unity, from Theorem 67 through Example 27 Content: This section starts with the computation of $\text{Gal}(GF(p^n)/GF(p))$, and then moves on to determine what we can say about Galois groups generated by adjoining roots of unity to some field. Comments: As in all sections, think carefully about the story you are telling in your section, and how it relates to things we've done and sections that follow yours. For instance, how does Theorem 69 relate to what we learn from Theorems 64 and 67?
Unit 3: Remote Instruction
In this unit we learn Galois theory while watching YouTube videos in our pajamas. Refresh this page before you watch any particular video, just to ensure that you're got the most current knowledge about known errata. Be as engaged with the material as you can be while watching the video: take notes; pause (the video) to think through anything that seems confusing; rewind when you realize something wasn't clear; have your text on hand to cross-reference assertions or citations; make a running list of issues that are confusing you; later, go through that list and see what issues you can resolve on your own; finally, you should feel very free to email me with any lingering issues that can't get resolved. Email me when you notice mistakes so I can post errata to this page.
-
Text section: The Galois Group Video: https://youtu.be/xhCeG-REtEw Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section11-II.pdf Content: We have already defined what the Galois group for an extension is, and we've seen that elements in the Galois group act by permuting roots of polynomials from the "ground field." But how does one go about determining how elements in the Galois group act on general field elements? We answer that question in this video. Errata: For slide 3 (discussed from 3:00-4:15), the second corollary should specify that the polynomial which E is the splitting field for is *separable*. (This issue is addressed in the posted slides.) -
Text section: The Galois Group Video: https://youtu.be/pty7kVKIRYM Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section11-III.pdf Content: We explicitly describe how elements in the Galois group can be defined by "climbing the tower" of intermediate fields. Errata: There's a significant mistake at 18:51 and 20:10. The theorem that lets us extend an isomorphism $\sigma:F \to F'$ up to an isomorphism for $F(\beta)$ prescribes that this extended map must send $\beta$ to a root of the image of its irreducible polynomial under $\sigma^*$ (in the video, I instead say that $\beta$ can go to any root of its irreducible polynomial, which is not correct). This doesn't change anything at the 18:51 mark, since the image of the irreducible polynomial for alpha over the rationals is unchanged under the identity function, but it does have a big effect on the "second layer" of this process (i.e., when trying to extend the isomorphism on $\mathbb{Q}(\alpha_1)$ to an isomorphism on $E$, since if $\psi$ is the isomorphism with domain $\mathbb{Q}(\alpha_1)$ we're trying to extend up to $E$, then $\alpha_2$ is supposed to go to some root of $$\psi(\text{irr}_{\mathbb{Q}(\alpha_1)}(\alpha_2)) = \psi(x^2+\alpha_1x+\alpha_1^2) = x^2+\psi(\alpha_1)x+\psi(\alpha_1)^2.$$ (This issue has been addressed in the posted slides.) -
Text section: The Galois Group Video: https://youtu.be/AzFMp8K6gDw Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section11-IV.pdf Content: In a tower of fields that includes a "base field," and "intermediate field," and a "top field," we discuss how the various Galois groups in play relate to each other. Errata: -
Text section: Roots of Unity Video: https://youtu.be/Z8oR8b8R84I Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section12-I.pdf Content: We introduce the idea of roots of unity and find two familiar examples where roots of unity were playing a key behind-the-scenes role. We argue that the set of nth roots of unity in any field are cyclic. Errata: -
Text section: Roots of Unity Video: https://youtu.be/CTOsbvFP0Gk Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section12-II.pdf Content: In this video we consider how finite fields can be viewed (and deeply understood) by using roots of unity. Errata: -
Text section: Roots of Unity Video: https://youtu.be/bepv9W2zqhA Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section12-III.pdf Content: We consider extensions that are the splitting field of a "binomial" $x^n-c$ with $c \in F$, first in the case where $c=1$. We compute the corresponding Galois groups. Errata: On slide 12, at 44:33 in the video, I write $|\text{Gal}(\mathbb{Q}(\omega_p)/\mathbb{Q})| = p-1 = U(\mathbb{Z}_p)$. The right side of this expression should instead be $|U(\mathbb{Z}_p)|$. On the last slide, at 1:10:48 in the video, I wrote $\mathbb{F}$ instead of $F$ -
Text section: Independence of Characters Video: https://youtu.be/1oNm8QrUC0Y Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section14-I.pdf Content: Up to this point we've thought about how we can assign a subgroup of the automorphisms of $E$ to any field extension $E/F$. This time we flip the script, starting with subsets of the automorphisms of $E$ and using them to create subfields of $E$. Errata: Towards the bottom of slide 5, at about 15:28 in the video, I write $F$ when I should have written $\mathbb{Q}$ (since in this example, the base field is $\mathbb{Q}$). -
Text section: Independence of Characters Video: https://youtu.be/5ECzsJ5cF1Q Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section14-II.pdf Content: Last time we thought about the fixed field associated to some subset $\text{Aut}(E)$, and we were able to prove some nice results. Can we get better results if we assume more about the subset from the automorphism group? In this video, we see that when we take a subgroup of the automorphism group, we get some very nice results about the fixed field operator: it "preserves bigness" and is injective. Errata: -
Text section: Independence of Characters Video: https://youtu.be/BYJauVHiQMk Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section14-III.pdf Content: In this video we develop the machinery necessary to prove that any finite collection of automorphisms of E are "E-independent". This independence was one of the key ingredients for us to show that fixed fields weren't too small. It's been separated from the rest of the content since it's not something Galois-theoretic per se. You can choose to not watch this video if you prefer. Errata: -
Text section: Galois Extensions Video: https://youtu.be/QJZP58IySSY Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section15-I.pdf Content: We saw in the Independence of Characters video that Galois groups can't be "too big," in that they are bounded by the degree of the fixed field extension. Under what circumstances do Galois groups achieve this maximal size? Using some ideas we've already seen, it turns out that this occurs precisely when $F$ is the fixed field of $\text{Gal}(E/F)$. We also know this occurs when $E$ is the splitting field for separable polynomials. Are there other conditions that are sufficient to make $\text{Gal}(E/F)$ as big as it can be? We give a few different ways to describe this phenomenon. Errata: -
Text section: Galois Extensions Video: https://youtu.be/BdzllYIg6CU Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section15-II.pdf Content: Now that we have defined what it means for an extension to be Galois, what can we say about "sub-Galois extensions"? From Assignment 9, problem 4 we know that if $E/F$ is Galois and $K$ is a field between $E$ and $F$, then $E/K$ is Galois as well. But when is $K/F$ Galois? In this lecture, we give a few equivalent characterizations of this phenomenon. Errata: -
Text section: The Fundamental Theorem of Galois Theory Video: https://youtu.be/zOol8J7aX_U Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section16-I.pdf Content: Building on all the results we've developed thus far, we wrap them all up into a single result that shows that the collection of subgroups of $\text{Gal}(E/F)$ and the collection of fields within $E/F$ have a strong relationship to each other --- at least when $E/F$ is itself Galois. Errata: -
Text section: The Fundamental Theorem of Galois Theory Video: https://youtu.be/4AuPkh2sXEk Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section16-II.pdf Content: We use the fundamental theorem of Galois theory to consider the relationship between the lattice of intermediate fields and the collection of subgroups of the Galois group for a few familiar field extensions. Errata: -
Text section: Applications Video: https://youtu.be/ydbOonBdl3M Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section17-I.pdf Content: We prove the fundamental theorem of algebra using Galois theory. Errata: -
Text section: Solvable by Radicals and Galois' Great Theorem (loosely inspired) Video: https://youtu.be/uRY8Jox0Ngo Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section18-I.pdf Content: In this video we give a definition that helps us conceptualize what it means for there to exist some formula "analogous to the quadratic, cubic and quartic" formulas which "solve for the roots" of a given polynomial. We given several examples of polynomials which satisfy this condition. We then discuss a (seemingly unrelated) group-theoretic condition called solvability, giving a handful of examples and non-examples. We then state Galois' Great Theorem, which says that a polynomial is "solvable by radicals" if and only if its corresponding Galois group (i.e., the Galois group of its splitting field) is a solvable group. Note: this video is LOOOONG. Plan to take a break at (at least) about the 56 minute mark.
Errata: -
Text section: Solvable by Radicals and Galois' Great Theorem (loosely inspired) Video: https://youtu.be/ulm7mfRqpgs Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section18-II.pdf Content: We give a proof of Galois' Great Theorem. In addition to using the fundamental theorem of Galois theory in important ways, we also make repeated use of Kummer theory. Note: this video is LOOOONG. Plan to take a break at (at least) about the X minute mark.
Errata: -
Text section: Galois' Great Theorem (loosely inspired) Video: https://youtu.be/bLxEIOlh9t8 Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Section18-III.pdf Content: We tie up the loose end in our proof of Galois' Great Theorem by proving "Part II" of Kummer theory. Errata: -
Text section: Appendix C (loosely inspired) Video: https://youtu.be/SiKDVTyRmFg Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/AppendixC-I.pdf Content: We give a quick recap of some of the motivation behind Euclid's geometric formulation, and we discuss some famous unresolved problems from Greek geometry. We then view these questions from an algebraic perspective, and use the field-theoretic tools we've developed thus far to show that 3 famous "ruler and compass" problems are not solvable. You might want to have a copy of Euclid's Elements on hand while you're watching the first part of this video. Errata: -
Text section: N/A Video: Login with your Wellesley credentials Slides: http://palmer.wellesley.edu/~aschultz/w20/math306/Slides/Finale.pdf Content: We discussed some natural questions that a person might ask about what we've learned about Galois theory so far and how it could be pushed in future directions. This included a quick discussion of infinite Galois theory, as well as some "big problems" in modern Galois theory. We wrapped up with a description of some of my research projects. Errata: