MASTER’S EXAM, ALGEBRA, JANUARY 2013
1. Find a 3 ×3 orthogonal matrix with all entries nonzero. (No partial credit.)
2. Find the dimension of the space of n×nmatrices with trace 0. Justify your
answer. (The trace of a square matrix is the sum of its diagonal entries.)
3. Let Vand Wbe finite-dimensional vector spaces, and let L:V→Wbe a
linear transformation. Prove that dim(ker(L)) + dim(image(L)) = dim(V).
4. Let p, q be primes. Show that a group of order pq is not simple.
5. Let Qbe the quaternion group of order 8. Prove that every subgroup of Qis
normal.
6. Let Gbe a group with center Z(G). Prove that if G/Z(G) is cyclic, then Gis
abelian.
7. Let Rbe a commutative ring with 1. Let x, y ∈Rand let Ibe the smallest
ideal containing xand y. Show that I={ax +by :a, b ∈R}.
8. In a principal ideal domain, show that every nonzero prime ideal is maximal.
9. Calculate the 36th cyclotomic polynomial Φ36(x).
10. Let Fbe a field with 125 elements. Does the polynomial x2+x+ 1 have a root
in F? (The polynomial should be considered as having coefficients in Z/5Z.)
Justify your answer.