Math 123 Problem set 1, due Monday, Feb. 5
Wednesday, January 31.
Problem 1. Assume an integral domain R contains a eld F as a subring,
and R is nitedimensional when regarded as a vector space over F.
Math 123 Problem set1 3,
due Wednesday, Feb. 21
Monday, February 12.
Problem 1. Factor x9 x and x27 x in F3 .
Problem 2. Factor x16 x in F4 and F8 .
Problem 3. Prove that if F is a nite eld and f (x)
Math 123 Problem set 4,
due Monday, Feb. 26
Wednesday, February 21.
Problem Find a primitive element for each of the following extensions:
1.
a) Q( 3, 5)/Q.
b) Q( 3, 5, 7)/Q( 7).
c) Q( 4 7, 3 2 + 1)/
Math 123 Problem set 2, due Monday, Feb. 12
Monday, February 5.
Problem 1. Prove the impossibility of constructing =
duplicating the cube problem).
3
2 (this is the
Problem 2.
a) Prove that the regula
Math 123 Takehome nal exam
due Wednesday, May 16.
Note 1: No collaboration is allowed on this exam: you cannot talk about
these problems with anyone. No references outside the course materials are
all
Math 123 Problem set 5,
due Monday, March 5
Monday, February 26.
Problem 1.
a) Let F be any eld (of characteristic zero) and let L = F (x1 , ., xn ) be
the eld of rational functions in n variables. De
Math 123 Problem set 6,
due Monday, March 19
Notation: R denotes a commutative ring with unit.
Monday, March 12.
Problem 1. Some examples.
a) Give an example of a module M over a ring R which is free
Math 123 Problem set 11,
due Friday, May 4.
Notation: An irreducible character is a character of an irreducible representation. If H G is a subgroup and is a character of H, we denote by
G the charact
Math 123 Problem set 10,
due Monday, April 23.
Notation: G is always a nite group, and any representation V of G is
nitedimensional as a vector space over C. When we say V is a representation of G, we
Math 123 Problem set 9,
due Monday, April 16.
Notation: G is always a nite group, and any representation V of G is
nitedimensional as a vector space over C. When we say V is a representation of G, we
Math 123 Problem set 7,
due Monday, April 2.
Monday, March 19.
Problem 1. Consider an abelian group A on four generators a, b, c, d,
with the following relations:
a) a + 2b + 7c 3d = 0, 3c 2d = 0, a +
Math 123 Problem set 8,
due Monday, April 9.
Problem 1.
a) Let V be the space of n n matrices with real entries. Prove that
(A, B) T r(At B)
is a positive denite symmetric bilinear form on V (do not c