Ph.D. Qualifying Exam in Algebra
BYU Department of Mathematics
Fall 2005
Do every problem. Give appropriately complete proofs for all of your answers. In your
solutions do not make assumptions or use facts which would make a problem trivial.
(1) Show that if pis a prime then every group of order pnhas a nontrivial center.
Recall that the center of a group is the set of group elements which commute with every group
element.
(2) Show that a finite simple group cannot have order 30.
(3) Show that every subgroup of a solvable group is solvable.
(4) Show that every vector space contains a basis.
(5) Show that if Ris a commutative ring, M a Noetherian R-module and u:M→Ma surjective
R-module homomorphism then uis an isomorphism. Recall that an R-module is Noetherian if
every strictly ascending chain of submodules is finite. Hint: Consider {ker(un)}.
(6) Show that the intersection of all proper nontrivial ideals of a commutative ring with unity is an
ideal in which every nonzero element is a zero-divisor. Hint: Show that this ideal is principal
and consider the square of each element.
(7) Show that if Fis a subfield of the finite field Kthen the Galois group of Kover Fis cyclic.
(8) Find a nonzero polynomial with coefficients in Zwhich has 3
√2−33
√4 + 1 as a root.
(9) Give an example of an nonseparable irreducible polynomial with coefficients in a field.
(10) Suppose Ris a commutative ring and f∈R[x] is a polynomial which has a root in R. Show
that R is Noetherian if and only if R[x]
(f(x)) is Noetherian.
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