07-401/Class Notes for April 11

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In Preparation

The information below is preliminary and cannot be trusted! (v)

The Fundamental Theorem of Galois Theory

It seems we will not have time to prove the Fundamental Theorem of Galois Theory in full. Thus this note is about what we will be missing. The statement appearing here, which is a weak version of the full theorem, is taken from Gallian's book and is meant to match our discussion in class. The proof is taken from Hungerford's book, except modified to fit our notations and conventions and simplified as per our weakened requirements.

Here and everywhere below our base field will be a field of characteristic 0.

Statement

Theorem. Let be a splitting field over . Then there is a correspondence between the set of intermediate field extensions lying between and and the set of subgroups of the Galois group of the original extension :

.

The bijection is given by mapping every intermediate extension to the subgroup of elements in that preserve ,

,

and reversely, by mapping every subgroup of to its fixed field :

.

Furthermore, this correspondence has the following further properties:

  1. It is inclusion-reversing: if then and if then .
  2. It is degree/index respecting: and .
  3. Splitting fields correspond to normal subgroups: If in is a splitting field then is normal in and .

Lemmas

The two lemmas below belong to earlier chapters but we skipped them in class.

The Primitive Element Theorem

The celebrated "Primitive Element Theorem" is just a lemma for us:

Lemma 1. Let and be algebraic elements of some extension of . Then there exists a single element of so that . (And so by induction, every finite extension of is "simple", meaning, is generated by a single element, called "a primitive element" for that extension).

Proof. See the proof of Theorem 21.6 on page 375 of Gallian's book.

Splitting Fields are Good at Splitting

Lemma 2. (Compare with Hungerford's Theorem 10.15 on page 355). If is a splitting field over and some irreducible polynomial has a root in , then splits in .

Proof. Let be a splitting field of over . We need to show that if is a root of in , then (so all the roots of are in and hence splits in ). Consider the two extensions

and .

The "smaller fields" and in these two extensions are isomorphic as they both arise by adding a root of the same irreducible polynomial () to the base field . The "larger fields" and in these two extensions are both the splitting fields of the same polynomial () over the respective "small fields", as is a splitting extension for and we can use the sub-lemma below. Thus by the uniqueness of splitting extensions, the isomorphism between and extends to an isomorphism between and , and in particular these two fields are isomorphic and so . Since all the degrees involved are finite it follows from the last equality and from that and therefore . Therefore .

Sub-lemma. If is a splitting extension of some polynomial and is an element of some larger extension of , then is also a splitting extension of .

Proof. Let be all the roots of in . Then they remain roots of in , and since completely splits already in , these are all the roots of in . So

,

and is obtained by adding all the roots of to .

Proof of The Fundamental Theorem

Proof of . More precisely, we need to show that if is an intermediate field between and , then . The inclusion is obvious, so we turn to prove the other inclusion.

Proof of .

Proof of Property 1.

Proof of Property 2.

Proof of Property 3.