MASTER’S EXAM, ALGEBRA, FEBRUARY 2011
1. If the columns of an orthogonal matrix are permuted, prove that the result is
still an orthogonal matrix.
2. Are the matrices
1 0 0
0 2 0
0 0 3
and
1 4 5
0 2 6
0 0 3
similar? Explain.
3. Let Ube a finite-dimensional vector space, and let Vand Wbe subspaces of U.
Give a formula relating the dimensions of V,W,V+W, and V∩W. Prove that
your formula is correct. (Note: You may use without proof the vector space
analogues of the homomorphism theorems from group theory.)
4. Show that there is no simple group of order 148.
5. Let Hbe a subgroup of finite index of an infinite group G. Prove that Ghas a
normal subgroup Kof finite index in Gwith K⊂H.
6. Determine the last 3 digits of the number 132011. Explain your method.
7. (a) Let Abe a commutative ring with 1. An element a∈Ais said to be
nilpotent if an= 0 for some positive integer n. Prove that the nilpotent
elements of Aform an ideal in A.
(b) Does the result of part (a) still hold if the hypothesis of commutativity is
dropped? Prove or disprove.
8. Let Rbe a commutative ring with 1. Prove that the principal ideal (x) in the
polynomial ring R[x] is a maximal ideal if and only if Ris a field.
9. Prove that the Galois group of the splitting field of x4−2 over Qhas order 8
and contains an element of order 4.
10. Let Fbe a field with 81 elements. Does the polynomial x2+1 have a root in this
field? (The polynomial should be considered as having coefficients in Z/3Z.)