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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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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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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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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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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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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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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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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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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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
p^{n} 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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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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We continued exploring finite fields. We first proved
that any finite field is isomorphic to

for some irreducible polynomial π in

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Today we completed the proof that for any prime power
p^{n} there exists a field with precisely
p^{n} 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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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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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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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.