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. Prove
that R is a eld.
Problem 2. If d2 are squarefree
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) F [x] is irreducible,
then f has no multiple roots in a
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)/Q.
d) K/Q, where K is a splitting eld of the polynomial
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 regular 9gon is not constructible by straightedge and
compass
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
allowed. In particular, you cannot go and try to nd books
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. Denote by si the i-th symmetric
polynomial in x1 , ., xn
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 (i.e., M
is isomorphic to Rn for some n) and a generati
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 character of G induced by .
Monday, April 23.
Problem 1.
a) Su
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 mean that V is a nitedimensional C-vector space togeth
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 mean that V is a nitedimensional C-vector space togethe
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 + 9c + d = 0;
b) a + b + 2c + d = 0, 2a + 3b + 4c = 0, 4
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 check absolutely
all the details).
b) What is the signat