Miami University The Additive Inverse and Polynomials Questions

Intro to Abstract Algebra

1. Let R be a finite integral domain with characteristic at least 3. Let 1 denote the unity of R.

(a) Prove that 1 ?= ?1, where ?1 denotes the additive inverse of 1.

(b) Show that 1 and ?1 are the only elements whose multiplicative inverse is itself. (c) Prove that the order of R is odd.

2. Let R be a ring and A,B ideals of R.

(a) Prove that A + B = {a + b : a ? A, b ? B} is an ideal.

(b) Define AB = {a1b1 +···+anbn : where n ? Z+ and for alli = 1,…,n that AB is an ideal of R.

3. Let Z2[x] be the ring of all polynomials with coefficients in Z2. (a) Show that Z2[x]/?x2 + x + 1? is a field.

(b) Show that Z3[x]/?x2 + x + 1? is not a field.

ai ? A,bi ? B}. Prove

(Comment: as discussed in class, the elements of Z2[x]/?x2 + x + 1? are ax + b + ?x2 + x + 1? over all a, b ? Z2. The most direct way to address (a) and (b) is to construct the multiplication table.)

4. Let n be an integer with decimal representation akak?1 · · · a1a0. Prove that n is divisible by 11 if and only if a0 ?a1 +a2 ?···+(?1)kak is divisible by 11.

5. (a) Let m, n be positive integers. Prove that an integer k can be written as mx + ny for some integers x and y if and only if k is a multiple of gcd(m, n).

(b) Let R be the ring of integers under addition and multiplication. Find a positive integer a such that ?a? = ?25?+?35?. Justify your work. Here for any x ? R, ?x? = {rx : r ? R} denotes the principal ideal generated by x.