Lecture 1 (Jan 31)
We discussed how to define numbers. This turned out to be remarkably subtle, as the case of the imaginary number i illustrated. This led us to the notion of algebraic distinguishability.
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Lecture 2 (Feb 5)
Today I presented Arnold's (non-Galois!) proof that there does not exist any quintic formula built out of finitiely many continuous functions and radicals. Click below for a write-up. (The write-up is intended for a general audience -- don't worry about the exercises.) You should also watch the 15-minute video which first exposed me to the proof, as well as play around with the the coefficient-roots tool we played with in class.
Click here for a write-up of Arnold's proof
Lecture 3 (Feb 8)
We started class by briefly discussing the insolvability of the quintic. In particular, I described what we'll be able to deduce from Galois theory, and compared and contrasted this with our conclusion from Arnold's approach. (We also showed that it's possible to solve any quintic using radicals... provided you allow infinite nesting of radicals.) Then, working together as a class, we figured out how to find the roots of the cubic x3-3x2-3x+7. Detailed notes below.
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Lecture 4 (Feb 12)
We began by reviewing some true facts about symmetric groups. After this we moved on to the main topic for today: a sketch of the Galois algorithm. We computed the Galois groups of two polynomials, and deduced what Galois theory would predict for the description of the roots. Detailed notes below.
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Lecture 5 (Feb 15)
We discussed the ring K[t] of polynomials with coefficients in a field K, in particular drawing an analogy between this ring and the integers. This led to a brief review of some ring theory. Then we played the Game of 15, which led to a discussion of what an isomorphism between two spaces means: that the two spaces are really the same, just expressed using different notation. Finally, we briefly explored how one might approach solving the equation x2+1=0 in the field of three elements. A detailed lecture summary is below.
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Lecture 6 (Feb 19)
We discussed the idea of generating a field from a given set. We then returned to our idea from last class about adjoining the imaginary number i to the field of three elements. This didn't work, because we proved the impossiblility of embedding this field into C. Our proof led us to formulate the notion of the characteristic of a field, as well as that of a field extension. Finally, we stated a fundamental theorem of Kronecker's about field extensions, and worked through an example of how his proof works. A detailed lecture summary is below.
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Lecture 7 (Feb 22)
We unpacked the proof of Kronecker's theorem, in particular working through the proof in a special case. We also saw that we need to be careful when we use it that we mod out by an irreducible polynomial. This led to a discussion of a few irreducibility tests. A detailed lecture summary is below.
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Lecture 8 (Feb 26)
Today we continued discussing irreducibility tests, in particular Eisenstein's criterion and local reduction (to a finite field). We also stated a beautiful result due to Schur. A detailed lecture summary is below.
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Lecture 9 (Mar 1)
After clarifying the importance of Kronecker's theorem, we flipped it on its head. Namely, given some number living in a field extension of a given field, how does one approach constructing the smallest field extension containing this number? Along the way we defined the notions of algebraic, transcendental, and minimal polynomial. A detailed lecture summary is below.
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Lecture 10 (Mar 5)
We continued our discussion of the degree of an extension, providing some new examples. Our last example, in particular, led to a general result: the Tower Law. (We also observed that the Tower Law looks remarkably like the third isomorphism theorem from group theory... which isn't a coincidence, as we'll see later.) We then gave a nice characterization of the minimal polynomial, and proved that adjoining an element α to a field K produces an extension whose degree is the degree of the minimal polynomial of α over K. We concluded with a brief discussion of the transcendence of e and π and the Hermite-Lindemann-Weierstrass theorem. A detailed lecture summary is below.
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Lecture 11 (Mar 8)
We introduced the notion of an algebraic extension, and proved a number of results related to algebraic extensions (most notably that any finite extension must be algebraic). Then we turned to a seemingly unrelated topic: compass and straightedge constructions. After coming up with numerous constructions as a class, I listed four constructions the Greeks never managed to accomplish. Using the theory of field extensions, we will prove these constructions to be impossible! A detailed lecture summary is below.
Click here for Lecture 11 notes
Lecture 12 (Mar 12)
Today we applied the theory of field extensions (following ideas of Pierre Wantzel) to settle three ancient questions: we proved the impossiblity of doubling the cube, trisecting the angle, and constructing the regular heptagon. (We also sketched the impossibility of squaring the circle.) Along the way we discussed the characterization of constructible regular polygons, in particular introducing the notion of a Fermat prime. A detailed lecture summary is below.
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Lecture 13 (Apr 5)
We started by observing that Kronecker's theorem, while always producing a root of a given polynomial, might not produce all its roots. This led us to define the notion of splitting field. We also drew a diagram of all the intermediate fields between Q and the splitting field in our example. We then found a nice group to associate to a given field: the automorphism group. We explored a few examples of automorphism groups. Finally, we drew a diagram of subgroups of the automorphism group to discover a remarkable correspondence: the Galois correspondence. A detailed lecture summary is below.
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Lecture 14 (Apr 9)
We started by discovering a way to produce many roots of a polynomial given just one of its roots. Remarkably, this method works even when we don't know the polynomial itself! This led us to formally define the notion of Galois conjugates, as well as of the automorphism group of an extension. We then returned to our discovery from last time about the correspondence between a diagram of intermediate fields and subgroups of a certain group. We realized the correspondence goes deeper than meets the eye, and this inspired us to state a formal theorem. This theorem turned out to be incorrect. We invented three different ways one might fix the theorem; these three ways all turned out to be equivalent. Any one of these three can be taken as the definition of a Galois extension. A detailed lecture summary is below.
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Lecture 15 (Apr 12)
We played around a bit more with Galois extensions, and observed that they possess both nice and annoying properties. We then stated the formal definition of a Galois group of an extension, as well as the definition of the Galois group of a polynomial. Then we stated the Fundamental Theorem of Galois theory. Although this is a very long and involved theorem, almost all of it is intuitive. We then presented a recent proof (published in 2014 by Geck) which bounds the size of the automorphism group of an arbitrary extension. In the case of a Galois extension, Geck's proof quickly implies part of the equivalence of the three alternative definitions for being Galois, as well as giving (without any special effort) the Primitive Element Theorem in the case of Galois extensions. A detailed lecture summary is below.
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Lecture 16 (Apr 16)
We completed (up to an exercise) the proof of the equivalence of the three definitions of an extension being Galois. Then we discussed separability, in particular discovering that one can determine separability of a polynomial without finding its roots. A detailed lecture summary is below.
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Lecture 17 (Apr 19)
We continued exploring separability, eventually proving (for example) that every irreducible over any characteristic 0 field must be separable. We then illustrated the power of the Fundamental Theorem of Galois Theory by using it to prove the Fundamental Theorem of Algebra. A detailed lecture summary is below.
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Lecture 18 (Apr 23)
We proved the first three parts of the Fundamental Theorem of Galois Theory. A detailed lecture summary is below.
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Lecture 19 (Apr 26)
We proved that applying an automorphism on the field side corresponds to conjugating by that automorphism on the group side under the Galois correspondence. Using this, we proved the fourth clause of the Fundamental Theorem of Galois Theory and sketched a proof of the fifth clause. Finally, we stated an important result about minimal polynomials, which we called the Fundamental Lemma. A detailed lecture summary is below.
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Lecture 20 (Apr 30)
After a brief review of the Fundamental Lemma, we introduced the concept of normal extensions. We then proved a new equivalent definition of Galoisity in terms of normality, and stated a criterion for normality. Next we explored two notions of unique factorization in the context of finite groups: the Krull-Schmidt theorem, and the Jordan-Holder theorem. A detailed lecture summary is below.
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Lecture 21 (May 3)
We continued our discussion of Jordan-Holder from last time. Then we introduced Galois' main ideas, in particular the notion of a radical extension. This led us to formulate a criterion for a polynomial to be solvable in radicals, which we then used to prove that a specific quintic polynomial can not be solved in radicals. We finished class by giving a heuristic (but incorrect) argument for Galois' solvability criterion. A detailed lecture summary is below.
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Lecture 22 (May 7)
Today we give a completely rigorous proof of Galois' solvability criterion (building on the heuristic but flawed argument from last class). Among other things, this necessitated a slight adjustment to our definition of a simple radical extension to allow roots of unity to be used in radical expressions. A detailed lecture summary is below.
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Lecture 23 (May 10)
We began by exploring an important question lingering from last class: must the splitting field of a solvable polynomial be a radical extension of Q? It turns out the answer is negative, and we gave an explicit example of such a polynomial (as well as proved a more general result about normal extensions of odd prime degree). We then discussed how to determine the Galois group of a given polynomial, working out a specific example as a vehicle for the discussion. A detailed lecture summary is below.
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Lecture 24 (May 14)
The bulk of the lecture focused on finite fields. Among other results, we proved that the multiplicative group of any finite field is cyclic. We then used this to characterize finite fields: for each prime power there exists precisely one (up to isomorphism) finite field with that many elements. We then developed the Galois theory of finite fields, which turned out to be quite simple (once we came up with the concept of the Frobenius automorphism). We next turned to some additional tricks for determining the Galois group of a given polynomial: the discriminant, a beautiful theorem of Dedekind, and a fantastic and under-appreciated theorem due to Frobenius. Finally we discussed cyclotomic extensions, tying up some loose ends from our exploration of constructibility earlier in the course. A detailed lecture summary is below.
