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Abstract Algebra by Prabhat Choudhary

By Prabhat Choudhary

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1' ... , ak)' Proposition If/ E F1x] and deg/= n, then/has a splitting field Kover Fwith [K: F] ::; n!. Proof We may assume that n ~ 1. ), F has an extension E 1 containing a root 0. 1 off, and the extension F( a I )/F has degree at most n. (Since j(a l ) = 0, the minimal polynomial of a l divides f) We may then write j(X) = (X - Ul)'1 g(x), where a l is not a root ofg and deg g ::; n - I. 2 of g, and the extension F(a 1• ( 2) will have degree at most n - lover F(a l ). Continue inductively and use to reach an extension of degree at most n!

8. If R is a UFD and P is a nonzero prime ideal of R, show that P contains a nonzero principal prime ideal. Principal Ideal Domains and Euclidean Domains A principal ideal domain is a unique factorization domain, and this exhibits a class of rings in which unique factorization occurs. We now study some properties of PID' s, and show that any integral domain in which the Euclidean algorithm works is a PID. If I is an ideal in Z, in fact if I is simply an additive subgroup of Z , then I consists of all multiples of some positive integer n.

1. If R is a UFO then R satisfies the ascending chain condition (acc) on principal (a2) ~ ... , then the eventuaIly stabilizes, that is, for some n we have (an+l) = (a n+2) = ideals: If a), a2, ... belong to Rand (al) ~ sequ~nce ~ M 2. If R satisfies the ascending chain condition on principal ideals, then R satisfies UFl, that is, every nonzero element of R can be factored into irreducibles. 3. If R satisfies UFl and in addition, every irreducible element of R is prime, then R is a UFO. Thus R is a UFO if and only if R satisfies the ascending chain condition on principal ideals and every irreducible element of R is prime.

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