Download e-book for kindle: Cyclic Galois Extensions of Commutative Rings by Cornelius Greither

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By Cornelius Greither

ISBN-10: 0387563504

ISBN-13: 9780387563503

ISBN-10: 3540563504

ISBN-13: 9783540563501

The constitution concept of abelian extensions of commutative jewelry is a subjectwhere commutative algebra and algebraic quantity thought overlap. This exposition is geared toward readers with a few heritage in both of those fields. Emphasis is given to the thought of an ordinary foundation, which permits one to view in a widely known conjecture in quantity thought (Leopoldt's conjecture) from a brand new attitude. tips on how to build definite extensions relatively explicitly also are defined at length.

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We want to m a k e it q u i t e c l e a r t h a t o u r a p p r o a c h h e r e is v e r y o l d - f a s h i o n e d , a n d t h a t t h e b e s t w a y o f u n d e r s t a n d i n g t h e K u m m e r s e q u e n c e is n o w a d a y s v i a c o h o m o l o g y . I t h a s been our wish to keep things very explicit, for later use. l~lotaUon-- a) F o r a n y c o m m u t a t i v e r i n g R, Pic(R) is t h e s e t o f i s o m o r p h i s m c l a s s e s o f i n v e r t i b l e R - m o d u l e s . Pic(R) is an a b e l i a n g r o u p , t h e c o m p o s i t i o n c o m i n g f r o m t e n s o r p r o d u c t o v e r R.

N o - l , w e g e t S t - . . ,,, Sno. Taking n = n 0, we g e t t h e f i r s t a s s e r t i o n o f b) f o r n = n o . W e s h o w [ S +t:S ~] = p by i n d u c t i o n over n > n o , t h e c a s e n = n o being s e t t l e d . The kernel D is e m b e d d e d via t0n+t in D', t h e k e r n e l o f ( Z / p ~ ÷ l ) * ( Z / p ~ ) *, a n d D' is t h e g r o u p o f o r d e r p g e n e r a t e d by t h e c l a s s o f l + p n m o d pn÷t. H e n c e [ S ÷1:S n] = p iff ton+t(Fn÷ 1) c o n t a i n s l + p n.

I}eflmltloL NB{R,G) denotes the set of all isomorphism classes of G-Galois extensions S/R which have a normal basis. By definition, NB(R,G) is a subset of H(R,G). 4. I f G is abelian, then N B ( R , G ) is a subgroup o f H ( R , G ) . Proof. 3 a n d c) j u s t a b o v e . To s i m p l i f y n o t a t i o n in t h e s e q u e l , w e i n t r o d u c e a n o t h e r n o t a t i o n : I ~ ~ For G abelian we define P(R,G) = H(R,G)/NB(R,G). R a m m ' k . O f c o u r s e , P(R,G) is c a n o n i c a l l y i s o m o r p h i c t o I m ( p i c ) c Pic(R[G]).

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Cyclic Galois Extensions of Commutative Rings by Cornelius Greither

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