Show that the set of elements in Q integral over Z is Q. (Hint: consider a fraction a/b in reduced form with b > 1 which

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Show That The Set Of Elements In Q Integral Over Z Is Q Hint Consider A Fraction A B In Reduced Form With B 1 Which 1 (110.5 KiB) Viewed 42 times

Show that the set of elements in Q integral over Z is Q. (Hint: consider a fraction a/b in reduced form with b > 1 which satisfies a monic relation with integer coefficients.) Following is some useful hints (Only need prove above, do not need prove below) Theorem 30.4 (Noether's Normalization Theorem). Let R be a finitely generated algebra over a field K. There exist elements u₁,..., ud ER which are algebraically independent and such that R is module-finite over K[u₁,..., ud]. Definition 29.1. Let R be a commutative ring and let S be an R-algebra. (a) We say that S is module finite over R if it is a finitely generated R-module. (b) If RC S, we say that an elements ES is integral over R if there exists a monic polynomial f(X) = Xd + ad-1Xd-¹ +...+ a₁X + ao € R[X] such that f(s) = 0. Proposition 29.2. Let R be a commutative ring. Let S be an R algebra and T be an S algebra. If S is module finite over R and T is module finite over S then T is an R-algebra which is module finite over R. Theorem 29.3. Let R be a commutative ring and let S be an R-algebra containing R. The following are equivalent: (a) S is module finite over R, (b) there exist 0₁,..., On ES which are integral over R such that S = R[0₁,..., On], (c) S is a finitely generated R-algebra and every element in S is integral over R.