Galois Groups as Groups of Permutations (Algebra 3: Lecture 26 Video 1)

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Lecture 26: We started this lecture by defining the Galois group of a separable polynomial of degree n f(x) in F[x]. We saw that Gal(f) is isomorphic to a subgroup of S_n. When f(x) is irreducible we saw that this subgroup is transitive and when f(x) is reducible we saw that we actually get a subgroup of a direct product of smaller symmetric groups. We considered two examples over Q corresponding to the polynomials (x^2-2)(x^2-3) and x^3-2. We defined elementary symmetric functions and defined the general polynomial of degree n over F in terms of its roots. We saw that S_n acts on F(x_1,...,x_n) and that the fixed field of this action is F(s_1,...,s_n). We saw how this led to the Fundamental Theorem of Symmetric Functions. We then discussed this kind of question again, but this time starting from the general polynomial of degree n defined in terms of its coefficients. We saw that either way we define this general polynomial we are led to the same place-- it is separable with Galois group S_n. In the last part of the video we focused on the discriminant of a polynomial. We proved that when char(F) is not equal to 2, Gal(f) is a subgroup of A_n if and only if Disc(f) is the square of an element in F. We saw how this discussion matched up with our earlier discussion of roots of quadratic polynomials.

Reading: In this lecture we very closely followed Section 14.6, pages 606-611. We mentioned the Fundamental Theorem of Symmetric Polynomials, which is discussed in Exercises 37-43 of Section 14.6. There is also a comment on page 609 in the paragraph leading up to Theorem 32-- "Conversely, if the roots of a polynomial f(x) are independent indeterminates over F, then so are the coefficients of f(x)-- cf. the beginning of Section 9." I find this discussion pretty confusing as written and recommend that you read the first few pages of Section 14.9 to clarify things a little.