North Idaho College Algebra Worksheet

DUE:(1) Consider the ring S = R[x]/(x¹+x²).
(a) Up to isomorphism, what are the cyclic modules of S?
(b) Which of these cyclic modules are projective?
(2) Consider the ring R = Z[√5] = {a+b√5 | a,b ≤ Z}. Define N: R→ Z by N(a+b√√5) =
a²562. It has the property N(aß) = N(a)N(3) for a, ß € R.
(a) Show that R has infinitely many units.
(b) Show that there is no a E R with N(a) = 2 (mod4).
(c) Show that the ideal (2,1 + √5) is not principal.
(d) Give an element of R that is irreducible, but not prime.
(3) Suppose that S is a commutative ring with identity, and R is a subring (that also contains
1 € S). We assume that S is finitely generated over R, i.e., there exist a₁, a2,…, an ES such
that every element of S is of the form p(a1, a2,…, an) for some polynomial p(x1, x2,…,xn)
with coefficients in R. Show that if R is noetherian, then S is noetherian.
(4) Suppose that Ui, i E I are Zariski open subsets of A” with Uier Ui = A”, where I is an index
set. Show that there exists a finite subset JCI such that U₁, i E J with Uiej U₁ = A”.
(5) A matrix A is called unipotent if A-I is nilpotent. Suppose that A E GLn(C) is an invertible
nxn matrix. Prove that there exist matrices A₁, A2 E GLn (C) such that A = A₁ A2 = A2A₁,
A₁ is diagonalizable, and A2 is unipotent. (Hint: First do the case where A is in Jordan
canoical form.)
(6) Suppose that V is a 3-dimensional real vector space and B =
{e1,e2, e3} is a basis of V.
Assume that : V → V is a linear map and that the matrix M(o) with respect to the
basis B is equal to
0 1 2
1 23
3 4 5
There exists a unique linear map 26: 1²V →
(v) A (w) for all v, w€ V.
(a) Give a basis of A2 V and give the matrix of
(b) What is the rational canonical form of ^²?
A2V with the property ^2p(vw):
2 with respect to this basis.
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