MASTER’S EXAM, ALGEBRA, FEBRUARY 2012
1. Show that the determinant of an orthogonal matrix must have absolute value 1.
2. Let Abe a nilpotent n×nreal matrix. (The matrix Ais said to be nilpotent if Ak= 0
for some positive integer k.) Show that the only possible eigenvalue of Ais 0.
3. Let P3be the set of polynomials with real coefficients of degree no more than 3. Show
that {1, x + 1, x2−x+ 1,1−x3}forms a basis for P3. Compute the matrix for the
linear map given by differentiation of polynomials with respect to that basis. Compute
a basis for the nullspace of the matrix.
4. Let Gbe a group, Ha subgroup of G, and a∈G. Let mbe the order of a, and let n
be the smallest positive integer such that an∈H. Prove that n|m.
5. Assume that every element of a group Ghas order ≤2. Prove that Gis abelian.
6. Show that any group of order 14 has a normal subgroup of order 7.
7. Let Rbe a finite commutative ring with 1 6= 0. Prove that if Rhas no zero divisors,
then Ris a field.
8. Let Fbe a field. Show that the ring F[x] of polynomials in one variable over Fis a
principal ideal domain.
9. Prove that if Eis a Galois extension of Qof odd degree, then E⊂R.
10. Show that if E/F is a finite extension of fields, then any subring Rof Econtaining F
is itself a field.