Extensions of Rings
It is not always desirable to deal only with field extensions. Sometimes one wants to obtain a field extension by reducing a ring extension modulo a prime ideal. This procedure occurs in several contexts, and so we are led to give the basic theory of Galo
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VII
Extensions of Rings
It is not always desirable to deal only with field extensions . Sometimes one wants to obtain a field extension by reducing a ring extension modulo a prime ideal. This procedure occurs in several contexts, and so we are led to give the basic theory of Galois automorphisms over rings , looking especially at how the Galois automorphisms operate on prime ideals or the residue class fields . The two examples given after Theorem 2.9 show the importance of working over rings, to get families of extensions in two very different contexts .
Throughout this chapter, A, B, C will denote commutative rings.
§1.
INTEGRAL RING EXTENSIONS
In Chapters V and VI we have studied algebraic extensions of fields . For a number of reasons, it is desirable to study algebraic extensions of rings . For instance, given a polynomial with integer coefficients, say X 5 - X-I, one can reduce this polynomial mod p for any prime p, and thus get a polynomial with coefficients in a finite field. As another example, consider the polynomial
where Sn-l' . . . , So are algebraically independent over a field k. This polynomial has coefficients in k[so, . . . , Sn-l] and by substituting elements of k for So, . . . , Sn -l one obtains a polynomial with coefficients in k. One can then get
333 S. Lang, Algebra © Springer Science+Business Media LLC 2002
334
EXTENSION OF RINGS
VII, §1
information about polynomials by taking a homomorphism of the ring in which they have their coefficients. This chapter is devoted to a brief description of the basic facts concerning polynomials over rings . Let M be an A-module. We say that M is faithful if , whenever a E A is such that aM = 0 , then a = O. We note that A is a faithful module over itself since A contains a unit element. Furthermore , if A 0, then a faithful module over A cannot be the O-module. Let A be a subring of B . Let a E B. The following conditions are equivalent:
*"
(NT I.
The element a is a root of a polynomial
X"
+ (/,, _I X,,-1 + ... + (/ 0
with coefficients a j E A, and degree n ~ I . (The essential thing here is that the leading coefficient is equal to I.)
(NT 2.
The subring ALa] is a finitely generated A-module.
(NT 3.
There exists a faithful module over A[a] which is a finitely generated A-module.
We prove the equivalence. Assume (NT I. Let g(X) be a polynomial in A[X] of degree ~ I with leading coefficient I such that g(a) = O. If I(X) E A[X] then I (X )
= q(X)g(X) + reX)
with q, r E A [ X ] and deg r < deg g. Hence I (a ) = rea), and we see that if deg g = II, then I , a, . . . , r:x ,,- I are generators of A [ ex] as a module over A . An equation g(X) = 0 with g as above, such that g(a) = 0 is called an integral equation for a over A. Assume (NT 2. We let the module be A[a] itself. Assume (NT 3, and let M be the faithful module over A [ ex] which is finitely generated over A , say by elements WI ' . .. , W no Since «M c M there exist elements aij E A such that
Transposing «wI' . . . , exlV" to the right-hand side of these equations, we conclude
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