Mathematics 352 – Abstract Algebra II Discussion 2 – More on Galois Groups March 28, 2008

Background

Given an extension field K of a field F, recall that we have defined

Gal_{F} K = {σ : K → K : σ is an F -automorphism of K. }

which is a group under the operation of function composition, called the Galois group of K over F. For instance, last Friday we saw that Gal_{ℚ}ℚ(\sqrt{2}, \sqrt{3}) = \{id, ρ, σ, τ\} is a noncyclic group of order 4, isomorphic to (ℤ_{2} × ℤ_{2}, +). Today we want to work out another example and start to understand a relationship between subgroups of Gal_{F} K and subfields of K containing F.

Discussion Questions

A) Consider the polynomial f(x) = x^{3} − 2 ∈ ℚ[x], and let K be the splitting field of f(x) over ℚ.

(1) Show that K = ℚ(\sqrt[3]{2}, ω), where ω = \frac{−1 + i\sqrt{3}}{2} is a cube root of 1.

(2) Consider the minimal polynomials f(x) = x^{3} − 2 of \sqrt[3]{2} and g(x) = x^{2} + x + 1 of ω in ℚ[x]. By the theorem we finished proving in class last Friday, we know that given any root v of f(x) and any root w of g(x), there must be some element ϕ of Gal_{ℚ}ℚ(\sqrt[3]{2}, ω) with ϕ(\sqrt[3]{2}) = v and ϕ(ω) = w. How many distinct elements of the Gal_{ℚ}ℚ(\sqrt[3]{2}, ω) are there? List them all.

(3) By numbering the 3 roots of x^{3} − 2 in some way, establish a 1-1 correspondence between Gal_{ℚ}ℚ(\sqrt[3]{2}, ω) and a subgroup of S_{3}. (Which permutation of the roots is induced by each element of the Galois group?) What is the structure of Gal_{ℚ}ℚ(\sqrt[3]{2}, ω)? (i.e. to which “standard” group is it isomorphic?)

B) Now we want to consider subgroups of Gal_{F} K and subfields of K.

(1) Show in general that if G = Gal_{F} K and H is a subgroup of G, then

E_{H} = {a ∈ K : σ(a) = a for all σ ∈ H}

is a subfield of K containing F (called the fixed field of H for a hopefully clear reason!). Note that this requires you to show that E_{H} is a subring of K which is also a field.

(2) Let F = ℚ and K = ℚ(\sqrt{2}, \sqrt{3}). The Galois group G = Gal_{ℚ}ℚ(\sqrt{2}, \sqrt{3}) has 5 different subgroups H (including the “trivial” subgroups {id} and G). What are the fixed fields of each of them?

(3) The Galois group G = Gal_{ℚ}ℚ(\sqrt[3]{2}, ω) has 6 different subgroups H (again including the trivial subgroups). Find the fixed fields of each one.

Assignment

Group writeups due in class on Monday, April 7.