Galois Theory: Fundamental Definitions and a Basic Example

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Évariste Galois has a tragic life story, though Galois Theory is amazingly powerful. First, given a field extension E of F, we need the definition of the Galois group Gal(E/F) of E over F. It is the group of all field automorphisms of E that keep all elements of F fixed. It is indeed a group. Also, given a subgroup H of Gal(E/F), the fixed field of H is E_H = {x ∈ E | φ(x)=x ∀ φ ∈ H}. This is indeed a subfield of E containing F. Galois Theory Example 1 is about Gal(ℚ(√2)/ℚ) = Gal(ℚ(2^(1/2))/ℚ). First, recall that ℚ(√2)={a + b√2 | a,b ∈ ℚ}, because √2 is a root of the polynomial x^2 - 2, which is irreducible over ℚ. Therefore, the degree of ℚ(√2) over ℚ is 2, that is [ℚ(√2):ℚ] = 2 = dimension of ℚ(√2) as a vector space over ℚ. Because any automorphism of a field containing ℚ must act as the identity on ℚ, this means and field automorphism φ of ℚ(√2) is completely determined by φ(√2), which itself must be a square root of 2. So, φ(√2) = ±√2. Ultimately, Gal(ℚ(√2)/ℚ) = {ε, φ}, where ε is the identity map and φ(a + b√2) = a - b√2, though we must also verify this is one-to-one, onto, doubly operation-preserving, and fixing ℚ. The subgroup lattice for Gal(ℚ(√2)/ℚ) matches the tower of fields (subfield lattice) for ℚ(√2) over ℚ, which matches the Fundamental Theorem of Galois Theory.

#GaloisTheory #GaloisGroup #FieldTheory

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I loved it. Wich book are you using for the lecture?

rathtrad

Oh dear, I struggled with abstract math's 20 years ago when doing my degree, nothing has changed since. I wish I could come to grips with it as I can see its value but it really hurts my brain :)

campbellmorrison

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