Lecture 1 (Sep 10)

We discussed administrative stuff. Then we started discussing a list of six problems we hope to resolve this semester using Galois theory.

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Lecture 2 (Sep 14)

We began by concluding our discussion of six motivational problems we hope to resolve during the course of the semester. We then switched to trying to solve the cubic equation. Although our initial attempts were unsuccessful, we were able to show that the cubic formula must contain both a square root and a cube root in it.

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Lecture 3 (Sep 17)

Brainstorming together, we were able to solve a cubic equation. Although a bit technically involved, the argument turned out to be not so hard, and required essentially one big insight: that we should guess a form of the roots which displays 3-fold symmetry.

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Lecture 4 (Sep 21)

We described the main algorithm of Galois theory, which assigns to any polynomial a group (consisting of all the symmetries of the polynomial) and then studies the structure of that group in order to deduce information about the shape of the roots of the polynomial. We worked out two examples, and sketched the proof of the impossibility of solving the quintic in radicals.

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Lecture 5 (Sep 24)

We introduced the nice notation e(x). We continued discussing the Galois algorithm. We reviewed permutation groups. We concluded with an analogy between the factorizations of integers and Jordan-Holder decomposition of groups, and a discussion of the Holder program.

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Lecture 6 (Sep 28)

We spent most of the lecture reviewing quotient groups. We also gave a brief account of Galois' turbulent life, and ended with a discussion of what it means for an equation to be unsolvable.

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Lectures 7-8 (Oct 1 and 5)

We discussed how to approach solving equations when solutions don't exist in the field you're working in. In particular, we developed the notions of characteristic and field extension. We then proved Kronecker's fundamental result, applied it to a few examples, and introduced the notion of the degree of an extension.

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Lecture 9 (Oct 8)

We explored the notion of the degree of a field extension. We proved that if f(x) is irreducible over a field F, then the degree of the extension K = F[x]/(f) over F agrees with the degree of the polynomial f. We also proved the Tower Law, which describes how the degree behaves on extensions of extensions. We then used the tower law to understand the extension Q(ω, ∛) over Q; this led us to draw a lattice of subfields. We finished with a brief discussion of criteria for irreducibility over Z.

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Lecture 10 (Oct 19)

Kronecker's theorem asserts that given a field F, any polynomial with coefficients in F has a root in some field extension of F. In this lecture, we flip this on its head: given an extension K / F and some element α in K, can we find a polymomial with coefficients in F which has α as a root? In the process of exploring this, we were led to define the notions of algebraic and transcendental elements, the minimal polynomial of a given element, and the degree of an element. We also proved that the sum (or product, or difference, or...) of two algebraic elements remains algebraic.

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Lecture 11 (Oct 22)

We concluded our discussion from last time by proving that every finite extension is algebraic. We then began exploring finite fields. We proved that every finite field has pn elements for some prime p and some positive integer n. We finished by constructing a field of four elements. Even this simple field has a structure which is unintuitive. Among other things, we noted that it's not cyclic with respect to addition, but is cyclic with respect to multiplication.

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Lecture 12 (Oct 26)

We first observed that a finite field F is cyclic with respect to addition iff the field has a prime number of elements. We then proved that the multiplicative group of any finite field is cyclic. We concluded by discussing several open problems related to finding a generator, for example, Artin's conjecture.

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Lecture 13 (Oct 29)

We continued exploring finite fields. We first proved that any finite field is isomorphic to

Fp[x] / (π)

for some irreducible polynomial π in Fp[x]. Continuing to explore properties of finite fields with pn elements, we were motivated to introduce the notion of splitting field, and proved that any finite field is a splitting field of a certain polynomial f(x) over Fp. This allowed us to mostly prove the existence of finite fields of arbitrary prime power size; the one assertion left unproved is that all roots of the polynomial f(x) are distinct in a field extension.

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Lecture 14 (Nov 2)

Today we completed the proof that for any prime power pn there exists a field with precisely pn elements. The key to the proof was developing the criterion for separability, allowing us to check whether or not a polynomial in F[x] has repeated roots in an extension without leaving the world of F.

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Lecture 18 (Nov 16)

Today we proved the Fundamental Theorem of Algebra, using the Fundamental Theorem of Galois Theory, some group theory, and a tiny bit of analysis (any polynomial of odd degree has a real root).

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Lecture 19 (Nov 19)

Today we proved an important by technical result which I am going to call the Isomorphism Extension Lemma. From this we were able to easily deduce that splitting fields are unique up to isomorphism, as well as obtaining a bound on the size of the automorphism group of an extension.

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Lecture 20 (Nov 23)

Today we gave an alternative (equivalent) definition of what it means for an extension to be Galois. The key was a new property an extension might have: normality. We explored these new notions, and ended by restating the Fundamental Theorem of Galois Theory in terms of this new terminology.

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