# Cyclic Galois Extensions of Commutative Rings by Cornelius Greither

By Cornelius Greither

The constitution idea of abelian extensions of commutative earrings is a subjectwhere commutative algebra and algebraic quantity thought overlap. This exposition is geared toward readers with a few historical past in both of those fields. Emphasis is given to the proposal of a typical foundation, which permits one to view in a widely known conjecture in quantity thought (Leopoldt's conjecture) from a brand new attitude. easy methods to build yes extensions rather explicitly also are defined at length.

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**Example text**

We will have t o t r e a t this in detail in C h a p t e r VI, and the m e t h o d s e m p l o y e d w o u l d lead us t o o far afield at the moment. Hence we just s t a t e the simplest case here. For a proof, see VI §1 or the references given there. S. Let p b e any p r i m e and R a ring o f characteristic p. ) g i v e n b y P ( x ) = x p - x , x ~ R . T h e n there is an i s o m o r p h i s m ~: R/PR ~> H(R,Cp) with j ( x ) = R [ Y ] / ( Y p - Y - x ) , the C v - a c t i o n b e i n g g i v e n b y a Y = Y + 1.

08 Zp--exte~o,,To c o n c l u d e this i n t r o d u c t o r y c h a p t e r , we briefly discuss e x t e n s i o n s with "Galois group" Zp, the additive g r o u p o f p - a d i c n u m b e r s , or, if one p r e f e r s , the inverse limit o f Z/p n for n -~ co. , s e q u e n c e s satisfying a c e r t a i n compatibility} o f e x t e n s i o n s with Galois g r o u p Cpn = (cyclic g r o u p o f o r d e r p~ }, one for each n e ~q. Fix a p r i m e n u m b e r p, and in each Cpn fix a g e n e r a t o r o n, Define e p i m o r p h i s m s ~n: Cpn+l -~ Cpn by s e t t i n g ~n{On+t) = On.

Y" f o l l o w s f r o m the above remark. For the p r o o f o f the other inclusion {which is nontrivial), let us s t a r t with A ~ H(R, Cp~) s u c h t h a t A' = Sn®RA has a normal basis over S n. The explanation in the above remark can be c o m p l e m e n t e d as follows: if y • A, and if l ® y g e n e r a tes a normal basis o f A', t h e n by the t h e o r y o f faithfully flat descent, y already generates a normal basis o f A. ) The point of the a r g u m e n t is hence t o find a g e n e r a t o r o f a normal basis "upstairs" which c o m e s f r o m "downstairs".