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 $F$ will be a field of characteristic 0.

Statement

Theorem. Let $E$ be a splitting field over $F$. Then there is a correspondence between the set $\{K:E/K/F\}$ of intermediate field extensions $K$ lying between $F$ and $E$ and the set $\{H:H<\operatorname {Gal} (E/F)\}$ of subgroups $H$ of the Galois group $\operatorname {Gal} (E/F)$ of the original extension $E/F$:

$\{K:E/K/F\}\quad \leftrightarrow \quad \{H:H<\operatorname {Gal} (E/F)\}$.

The bijection is given by mapping every intermediate extension $K$ to the subgroup $\operatorname {Gal} (E/K)$ of elements in $\operatorname {Gal} (E/F)$ that preserve $K$,

$K\mapsto \operatorname {Gal} (E/K)$,

and reversely, by mapping every subgroup $H$ of $\operatorname {Gal} (E/F)$ to its fixed field $E_{H}$:

$H\mapsto E_{H}$.

Furthermore, this correspondence has the following further properties:

1. It is inclusion-reversing: if $H_{1}\subset H_{2}$ then $E_{H_{1}}\supset E_{H_{2}}$ and if $K_{1}\subset K_{2}$ then $\operatorname {Gal} (E/K_{1})>\operatorname {Gal} (E/K_{1})$.
2. It is degree/index respecting: $[E:K]=|\operatorname {Gal} (E/K)|$ and $[K:F]=[\operatorname {Gal} (E/F):\operatorname {Gal} (E/K)]$.
3. Splitting fields correspond to normal subgroups: If $K$ in $E/K/F$ is a splitting field then $\operatorname {Gal} (E/K)$ is normal in $\operatorname {Gal} (E/F)$ and $\operatorname {Gal} (K/F)\cong \operatorname {Gal} (E/F)/\operatorname {Gal} (E/K)$.

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. Let $a$ and $b$ be algebraic elements of some extension $E$ of $F$. Then there exists a single element $c$ of $E$ so that $F(a,b)=F(c)$. (And so by induction, every finite extension of $E$ 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.