M ATH 503 H OMEWORK 6, S PRING 2014
W EEKS OF F EBRUARY 24; DUE T HURSDAY M ARCH 6
1. Compute the character table of S4 and determine which ones of the irreducible characters of S4
are induced from a proper subgroup?
2. Let p be a prime number. Let H(Fp )
M ATH 503 H OMEWORK 5, S PRING 2014
W EEKS OF F EBRUARY 17; DUE T HURSDAY F EBRUARY 27
The base eld k of every nite dimensional linear representation is algebraically closed of characteristic 0 unless otherwise specied.
1.
(a) Compute the character table
M ATH 503 H OMEWORK 4, S PRING 2014
W EEKS OF F EBRUARY 10; DUE T HURSDAY F EBRUARY 20
1. Let G be a nite group, let S be a nite set and let : G S S be an action of G on S. Let V
be the space of C-valued functions on S. Let : G GLC (V ) be the linear repr
M ATH 503 H OMEWORK 2, S PRING 2014
W EEK OF JANUARY 27; DUE T HURSDAY F EBRUARY 6
1. Let R be a commutative ring and let G be nite group.
(a) Show that the tensor product R Z Z[G] has a natural structure as a ring. (You need to
dene the ring structure an
M ATH 503 H OMEWORK 1, S PRING 2014
W EEKS OF JANUARY 15 AND JANUARY 20; DUE T HURSDAY JANUARY 30
1. Let V,W be modules over a commutative ring R. Prove that there exists a unique R-linear isomorphism
s : V R W W R V
such that s(v w) = w v for all v V and
M ATH 503 H OMEWORK 3, S PRING 2014
W EEKS OF F EBRUARY 3; DUE T HURSDAY F EBRUARY 13
1. (Discriminant of a polynomial) Let x1 , . . . , xn be variables and let s1 , . . . , sn be the elementary
symmetric polynomials in x1 , . . . , xn . Let
n
F(t) := (t
M ATH 503 H OMEWORK 11, S PRING 2014
W EEK OF A PRIL 14; DUE T HURSDAY A PRIL 24
1. Give an examle of three elds F K L such that K/F and L/K are both nite Galois, but
L/F is not Galois.
2. Let M/F be a nite Galois extension of elds. Let be an element of o
M ATH 503 H OMEWORK 7, S PRING 2014
W EEK OF M ARCH 18; DUE T HURSDAY M ARCH 27
1. For every n N1 , dene the n-th cyclotomic polynomial n (T ) C[T ] by
n (T ) =
1/n
(T e2a
).
1an
gcd(a,n)=1
For instance 1 (T ) = T 1, 2 (T ) = T + 1, 3 (T ) = T 2 + T + 1,
M ATH 503 H OMEWORK 10, S PRING 2014
W EEK OF A PRIL 7; DUE T HURSDAY A PRIL 17
1. Determine the Galois group of the splitting eld of the polynomial T 4 3 over Q.
2. Let F = F7 (T ) be the fraction eld of the polynomial ring over F7 in a variable T .
(a)
M ATH 503 H OMEWORK 9, S PRING 2014
W EEK OF M ARCH 31; DUE T HURSDAY A PRIL 10
1. One of the problems in HW3 is to compute an explicit formula for the determinant of the a cubic
polynomial. The goal here is to tie the discriminant of a cubic polynomial w