Lecture 1 (Jan 31)
We saw that it's difficult to come up with a definition of i that doesn't simultaneously define -i, and made this precise by proving that there's no polynomial with real coefficients that has one as a root but not the other. Similarly, we proved that √2 and -√2 are algebraically indistinguishable, in the sense that there's no polynomial in ℚ[x] that has one as a root but not the other; we were forced to come up with a different proof of this. We ended with a discussion of the complex cube roots of unity ω and ω2.
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Lecture 2 (Feb 5)
We continued discussing algebraic indistinguishability, in particular proving that the three cube roots of 2 are indistinguishable over ℚ. However, one step in our proof wasn't rigorous: that 1, the real cube root of 2, and the square of the real cube root of 2 are `linearly independent' over ℚ. You will prove this on Problem Set 1. Next, we sketched a proof of Arnold's proof of the Abel-Ruffini theorem that there doesn't exist a quintic formula. A detailed write-up of this is on the assignments page.
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Lecture 3 (Feb 8)
We first briefly discussed the distinctions between what Arnold proves about the quintic and what we will prove using Galois theory about the quintic; in particular, we saw that each theory has advantages and disadvantages relative to the other. We next turned to a classical problem: finding the roots of a cubic polynomial. Ultimately, our key insight was that if we can guess the shape of the roots, then finding the roots isn't so challenging.
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Lecture 4 (Feb 12)
We began with a review of the group of permutation Sn and some of its structure. Then we gave a heuristic (and highly non-rigorous!) overview of one of the tools we'll develop over the course of the semester: an algorithm that, given a polynomial in ℚ[x], predicts what the roots look like (not what it is, but what it looks like when written down).
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Lecture 5 (Feb 15)
We reviewed some basics from ring and field theory, in particular the notions of irreducibility, unit, subring, and ideal.
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Lecture 6 (Feb 19)
Last time we discussed irreducible polynomials. This concept is highly reminiscent of prime numbers, and we pursued this analogy between ℤ and K[t], where K is an arbitrary field. In particular, concepts like prime factorization, divisibility, and the quotient-remainder theorem all carry over. Moreover, one fundamental property of both rings is that any ideal you can build out of multiple elements can also be built out of a single element. The second part of the class was motivated by trying to solve the equation x2 + 1 = 0 over the three element field F3. One natural idea is to adjoin the imaginary number i to this field, but we figured out this was impossible to do because F3 doesn't embed into ℂ. (We proved this by inventing the notion of the characteristic of a field.) We concluded by stating a theorem of Kronecker about field extensions that will allow us to solve polynomial equations over arbitrary fields.
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Lecture 7 (Feb 22)
We started by restating Kronecker's theorem from last time. Prior to proving it, we applied it to a concrete example: generating a field extension of F3 in which x2+1 has a root. It turns out this field has nine elements, and we wrote out an explicit multiplication table for this field. We then adapted our methods to prove Kronecker's theorem in general.
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Lecture 8 (Feb 26)
We discussed six tools that are useful for determining whether a given polynomial in ℚ[x] is irreducible.
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Lecture 9 (Feb 29)
We discussed a final irreducibility test---Cohn's criterion---that makes explicit a connection between prime numbers and irreducible polynomials. Next we discussed some notations, old and new. Finally, we flipped Kronecker's theorem on its head: given a field K and some α that lives in some extension of K, can we express K(α) in the form K[t] / (f) for some polynomial f in K[t]? And how can we determine a suitable polynomial f?
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Lecture 10 (Mar 4)
We continued our exploration of a reverse Kronecker theorem. This led us to the notions of algebraic and transcendental numbers. We concluded by defining the degree of a field extension.
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