Notes for 4H Galois Theory 2002–3 Andrew Baker [05/02/2003]
Department of Mathematics, University of Glasgow. E-mail address:
[email protected] URL: http://www.maths.gla.ac.uk/∼ajb
√ 3 Q( 2, ζ3 ) g g g
√ Q( 3 2) N
I II ggggg mm II ggggg mmmmm II g g g g 2 g m II g m g ggg mm 2 II g 2 g m g g II 3 √ √ gggg II 3 3 2 II Q( 2 ζ ) Q( 2 ζ ) 3 3 II PPP DD N J II PPP D II D PPP D II DD PPP II DD PPP DD 3 PPP3 DD PPP 3 Q(ζ3 ) DD PPP Q DD PPP jjjj j j j D 2 jjj PPP j PPP DDD j jjj PPP DD jjjj PP jjjj
QS
¯
Gal(E/Q) ∼ = S3
³
nn z nnn zzz n n nn zz nnn zz 3 nnnnn 3 zzz 3 z nnn zz nnn z n n z z nnn zz nnn z n n nn ³ zz ·
{id, (2 3)} XXX {id, (1 3)}
{id, (1
XXXXX QQQ XXXXX QQQ XXXXX 2 XXXXX QQQQQQ 2 2XXXXXX QQQ ¸ XXX
TTTT TTTT2 TTTT TTTT °
{id, (1 2 3), (1 3 2)}
u uu uu u u uu 3 uuuu u 2)} uu uu u u uu uu u u
{id}
√ The Galois Correspondence for Q( 3 2, ζ3 )/Q
ii
Introduction: What is Galois Theory? Much of early algebra centred around the search for explicit formulæ for roots of polynomial equations in one or more unknowns. The solution of linear and quadratic equatio