Assignment 2, Math 3EE3
Due Feb. 5 in class
(1) Fix a ring R and let S be the set of all functions from R to R.
(a) If we put addition on S as follows: for f, g S
(f + g)(r) = f (r) + g(r) for all r R
and multiplication dened by multiplication of function
Math 3EE3, Test 2
Bradd Hart, Mar. 17, 2015
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full
Notes
Integer Exponents
We will start off this chapter by looking at integer exponents. In fact, we will initially
assume that the exponents are positive as well. We will look at zero and negative
exponents in a bit.
recall the definition of exponentiatio
Assignment 3, Math 3EE3
Due Feb. 26 in class
(1) Prove the division algorithm for polynomials over an arbitrary eld. That is, show
that if F is a eld and f, g F [x] then there are unique q, r F [x] such that
g = qf + r and deg(r) < deg(f ). Hint: prove th
Math 3EE3, Test 1
Bradd Hart, Feb. 10, 2015
Please write complete answers to all of the questions in the test booklet provided. Partial credit may be given for your work. Unless otherwise noted, you
need to justify your solutions in order to receive full
Assignment 5, Math 3EE3
Due Apr. 2, in class
1. Here is a problem that involves Zorns Lemma and algebraic closures:
we will prove that if F is a eld then any two algebraic closures (algebraically closed elds which are algebraic over F ) are isomorphic by
Assignment 4, Math 3EE3
Due Mar. 12 in class
(1) Suppose that p is prime and : Z[x] Zp [x] is dened by
computing the coecients from a polynomial in Z[x] mod p.
(a) Prove that is a homomorphism.
(b) Show that if f Z[x] and (f ) have the same degree then
if
Assignment 1, Math 3EE3
Due Jan. 20 in class
(1) Suppose R is a ring. We say that p R is a projection if p2 = p.
Show that if p is a projection then pRp = cfw_pap : a R is a
subring of R for which p is a multiplicative identity. Moreover,
show that if S i