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 1.
 Let G be a finite group containing a subgroup
H of index p, where p is the smallest prime divisor of G.
Prove that H is a normal subgroup of G.
 2.
 Prove that every group of order 15 is cyclic.
 3.
 Prove that there is no simple group of order 36.
 4.
 Prove that Euclidean domains are principal.
 5.
 Prove that the maximal ideals of the
polynomial ring
are in bijective correspondence with
complex numbers.
 6.
 Verify that
is irreducible.
 7.

be a field of order p^{n}, p a prime.
Show that the automorphisms of
form a cyclic group of order n.
 8.
 Determine the irreducible polynomial for
over
.
 9.
 Construct the finite field
of order 9, and find a generator
for the multiplicative group of nonzero elements of
.
Mark S. Gockenbach
20091102