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 g^{n} = e for all .
Prove that the multiplicative group of a finite field is
cyclic.
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=a_{1}.
There exists an infinite set of
nonunits of R such that b_{1}b_{2} ... b_{n} 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) = x^{n}-1 and let K be a splitting field for f(x)
over . Prove that the Galois group is
abelian.
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 .