07-401/Class Notes for April 11

07-401/Navigation Panel   # Week of... Links 1 Jan 10 About, Notes, HW1 2 Jan 17 HW2, Notes 3 Jan 24 HW3, Photo, Notes 4 Jan 31 HW4, Notes 5 Feb 7 HW5, Notes 6 Feb 14 On TT, Notes R Feb 21 Reading week 7 Feb 28 Term Test 8 Mar 7 HW6, Notes 9 Mar 14 HW7, Notes 10 Mar 21 HW8, E8, Notes 11 Mar 28 HW9, Notes 12 Apr 4 HW10, Notes 13 Apr 11 Notes, PM S Apr 16-20 Study Period F Apr 24 Final Add your name / see who's in! Register of Good Deeds

In Preparation

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

The goal of today's class is to prove (a weak but strong enough) form of the Fundamental Theorem of Galois Theory as follows:

Theorem. Let $F$ be a field of characteristic 0 and 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 $\operatorname {Gal} (K/F)\cong \operatorname {Gal} (E/F)/\operatorname {Gal} (E/K)$.