Problem Set #10
Due Mon., Nov. 26, 2012
1. Let n be an integer, and let 1 , . . . , n+1 be distinct real numbers. Let Pn R[x]
be the vector space of polynomials of degree n. Dene F : Pn Rn+1 by f
(f (1 ), . . . , f (n+1 ).
a) Show that F is an i
Problem Set #9
Due Mon., Nov. 19, 2012
1. Let I1 , . . . , In R be ideals in a commutative ring R.
a) Show that j =1 Ij = j =1 Ij if the ideals are pairwise relatively prime. Explain
this assertion geometrically in the case R = C[x, y ].
Problem Set #8
Due Mon., Nov. 12, 2012
1. Let p > 2 be a prime number, and let f (x) = x
a) Show that every square in (Z/p) is a root of f (x) (Z/p)[x].
b) Deduce that f (x) = i=1 (x ai ), where f is as in (a) and where cfw_a1 , . . .
Problem Set #7
Due Wed., Nov. 7, 2012
1. Prove, or disprove and salvage: If K is a eld, and f (x) K [x] has no roots, then
K [x]/(f (x) is a eld.
2. For each positive integer n, let Un = (Z/n) , the group of units modulo n. Find a generator of U1
Problem Set #6
Due Fri., Nov. 2, 2012
1. Which of the following are rings? (Note: To be a ring, it must have a multiplicative
identity.) For those which are not, why not? For those which are, are they commutative?
integral domains? elds?
Problem Set #5
Due Fri., Oct. 26, 2012
1. Which of the following groups are isomorphic: C2 C2 , Q, D4 , C2 ?
2. Find all groups of order 66, up to isomorphism. Which are simple? solvable? nilpotent?
abelian? cyclic? Which are split extensions (
Problem Set #4
Due Wed., Oct. 10, 2012
1. Let G be a p-group, and let be its Frattini subgroup.
a) Show that if g G then g p . (Hint: If H G is a maximal subgroup, show
that g p H by considering its image in G/H .)
b) Deduce that every element of
Problem Set #3
Due Mon., Oct. 1, 2012
1. a) How many ways can a regular tetrahedron be inscribed in a cube? (Here inscribing
is required to be tight. That is, the vertices of the tetrahedron are required to be placed
at vertices of the cube, and
Problem Set #2
Due Mon., Sept. 24, 2012
1. Let G be the symmetry group of a regular polyhedron P , let v be a vertex of P , and
consider the stabilizer H = cfw_g G | g (v ) = v of v . Must H be a subgroup of G? Must it
be a normal subgroup? In e
Problem Set #1
Due Wed., Sept. 19, 2012
1. Dene the center of a group G to be Z = cfw_g G | (h G) gh = hg .
a) Is Z a subgroup? Is it normal?
b) Find the center of Cn , Dn , Sn , An , Q, Z, GL2 (R).
2. If H is a subgroup of G, dene the normalizer
Problem Set #11
Due Mon., Dec. 3, 2012
1. Let R be the ring of polynomial functions on the unit sphere S 2 R3 . Thus this ring
is given by R = R[x, y, z ]/(x2 + y 2 + z 2 1).
a) Let P = (0, 0, 1) S 2 , and let RP = cfw_ f | f, g R; g (P ) = 0. Sh
For each of the following, either give an example, or else prove that none exists.
1. A non-abelian group of order 55.
2. A non-abelian group of order 121.
3. A simple group of order 256.
4. A nite group that is solvable but