(January 14, 2009)
[01.1] Let D be an integer that is not the square of an integer. Prove that there is no D in Q.
Suppose that a, b were integers (b = 0) such that (a/b)2 = D. The fact/principle we intend to invoke here
is that fractions can be put in lo
(January 14, 2009)
[04.1] (Lagrange interpolation) Let 1 , . . . , n be distinct elements in a eld k, and let 1 , . . . , n be any
elements of k . Prove that there is a unique polynomial P (x) of degree < n in k [x] such that, for all indices i,
P (i ) =
(January 14, 2009)
[03.1] Let R = Z/13 and S = Z/221. Show that the map
f :RS
dened by f (n) = 170 n is well-dened and is a ring homomorphism. (Observe that it does not map 1 R
to 1 S .)
The point is that 170 = 1 mod 13 and 170 = 17 10 = 0 mod 17, and 221
(January 14, 2009)
[02.1] Let G, H be nite groups with relatively prime orders. Show that any group homomorphism
f : G H is necessarily trivial (that is, sends every element of G to the identity in H .)
The isomorphism theorem implies that
|G| = | ker f |
(January 14, 2009)
[06.1] Given a 3-by-3 matrix M with integer entries, nd A, B integer 3-by-3 matrices with determinant
1 such that AM B is diagonal.
Lets give an algorithmic, rather than existential, argument this time, saving the existential argument f
(January 14, 2009)
[07.1] Classify the conjugacy classes in Sn (the symmetric group of bijections of cfw_1, . . . , n to itself).
Given g Sn , the cyclic subgroup g generated by g certainly acts on X = cfw_1, . . . , n and therefore
decomposes X into orbi
(January 14, 2009)
[11.1] Let be a primitive nth root of unity in a eld of characteristic 0. Let M be the n-by-n matrix with
ij th entry ij . Find the multiplicative inverse of M .
Some experimentation (and an exercise from the previous week) might eventu
(January 14, 2009)
[08.1] Let R be a principal ideal domain. Let I be a non-zero prime ideal in R. Show that I is maximal.
Suppose that I were strictly contained in an ideal J . Let I = Rx and J = Ry , since R is a PID. Then x is
a multiple of y , say x =
(January 14, 2009)
[12.1] Prove that a prime p such that p = 1 mod 3 factors properly as p = ab in Z[], where is a primitive
cube root of unity. (Hint: If p were prime in
Z
Z[], then Z[]/p would be a integral domain.)
The hypothesis on p implies that ( /p
(January 14, 2009)
[09.1] Show that a nite integral domain is necessarily a eld.
Let R be the integral domain. The integral domain property can be immediately paraphrased as that for
0 = x R the map y xy has trivial kernel (as R-module map of R to itself,
(January 14, 2009)
[10.1] Prove that a nite division ring D (a not-necessarily commutative ring with 1 in which any non-zero
element has a multiplicative inverse) is commutative. (This is due to Wedderburn.) (Hint: Check that the
center k of D is a eld, s