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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_1\subset K_2} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/K_1)>\operatorname{Gal}(E/K_1)} .
2. It is degree/index respecting: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [E:K]=|\operatorname{Gal}(E/K)|} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)]} .
3. Splitting fields correspond to normal subgroups: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E/K/F} is a splitting field then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(E/K)} is normal in $\operatorname {Gal} (E/F)$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K)} .