Fundamental butting ::i::::: Galois ness ) intermediate Chandu - PowerPoint PPT Presentation

Theorem Fundamental

butting ::÷÷÷÷÷÷÷÷i÷÷÷÷:÷:::: Galois ness ) intermediate Chandu Eakins of Thur ( Equivalent ¥ ( iii ) Gal ( Elk ) A Gal ( EIF ) ( iv ) Khs Galois . is

Leatherback HH et Aut ( E ) sub extensions of E subgroups . contra variance Properties " bigness " • preserves injective NcwSt " ? these " the Are same universes

Nn/BasicAenl - Gal ( E IF ) let G and Galois , E) F - be let subgroups of G set of be the . sub (G) let intermediate fields in EIF of be The set . hat ( Elf ) Let E 't FIH ) Sub (G) → Lat LEIF ) be - Define F : - ACK ) - Gul ( Elk ) : lat LEIF ) → Sub (G) be A - and F s ← O → G Lat IE IFI

Galois theory ) ( Fundamental Theorem of Thin - ids.ba , The and I satisfy A. F F functions - contravariant . both - ideate , e ) F. H , and and are - have Furthermore , we - 1¥ , : F ) [ FIH ) " for all HESUBCG ) , then KELAHEIF ) , then ¥aI , E Ck : F ) Iii ) for all Galois ift Ciii ) KIF Alk ) of is iff FIH ) Ip Civ ) Galois . HOG is

If we've contravariant , F already seen is contravariant get A 4 is problem hwk 9 we . by and , be HEG let - id sides fo F let's show New , - . we'll ( I. FXH ) = H show and given , . " ) 't ) - Gul ( ELE : GI E do FXH ) =L ( FCH ) ) - ( Nate . re Gul LE IE 't ) TEH has certainly true that any Its re Aut LE ) , and since implies TEH since • fixes EH . he H ) = { et E we got all EH for : h ( e ) - e - 't ) Gal ( E IE get HE So we .

other hand , the On " ) I : E " ) = IHI = [ E I Gal LEIE 1 EIEH is Galois since - GAKEIEH ) H we get finite , - ( Gal ( EIEH ) I is Since - ( A. F) CH ) ' - idiot CEIF ) F 'S FEKEE let show Now , let 's . . ⇐ " Elk ) = FLICK ) ) - F ( Gulf Elk ) ) ( Fo A) ( K ) = E' Then - EGAKEIK ) Since Elk is Galois , we get - K . .

for [ and Elf Galois G - Gall Elf ) since Now ' ¥ - " " - hit :b . - . . , - [ FIH ) ? F ) - . Ci " - Ck :b - = - . . , characterizations Equivalent content of " the ( iii ) & Civ ) are INGI " intermediate Guleismts of -