Algebra Prelim, Fall 1986
- Let G be a finite abelian (multiplicative) group. Prove
that if G is not cyclic, then there exists a positive integer n
such that n< |G| and gn = e for all .
- Prove that the multiplicative group of a finite field is
- Let G be a group of order , and let H
be a subgroup of order .
- Prove that H is abelian.
- Prove that H is normal in G.
- Prove that G is abelian.
- Let R be an integral domain. For an element ,
prove the equivalence of the following two statements.
- There exists an infinite chain of principal ideals of R with a=a1.
- There exists an infinite set of
nonunits of R such that b1 b2 ... bn divides a for each
positive integer n.
- Let R be a commutative ring with identity and let M be a
maximal ideal of R.
- Prove that .
- Conclude that M[x] is a prime ideal but not a maximal ideal
in R[x]. Indeed argue that there are infinitely many prime ideals
of R[x] which contain M[x].
- Let be fields such that K is a
splitting field over k. If , prove that
- Let K/k be a finite extension and let with
. If , prove that n | [K:k].
- Let f(x) = xn-1 and let K be a splitting field for f(x)
over . Prove that the Galois group is
- Let M be a module. A submodule S of M is small if
whenever S+N= M for any submodule N of M, then N=M. Suppose
S is small in M and there exists an epimorphism
where P is projective. Prove that there exists an epimorphism .
Wed Jul 31 15:25:22 EDT 1996