Math 150B Homework 3
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Math 150B Homework 3
1) Do problems 4.3, 4.4, 5.3, 5.5 from Chapter 11 of Artin (Note: if you don’t have access to the correct edition of Artin, please let me know or check with afellow student that has a current edition). For 5.3, is this a finite ring? How many elements? What is its structure?
2) True or false? Give a one sentence justification of each one.
a) The cancellation law holds in any ring that is isomorphic to an integral domain.
b) Every integral domain of characteristic 0 is infinite.
c) The direct product of two integral domains is an integral domain.
d) A zero divisor in a ring can have no multiplicative inverse.
3) Show that the characteristic of an integral domain must be either 0 or a prime p.
4) Prove that if p is a prime in an integral domain D, then p is irreducible. Is the converse true?
5) Factor the polynomial 4x2 − 4x + 8 into a product of irreducibles viewing it as an element in
a) the integral domain Z[x]
b) the integral domain Q[x]
c) the integral domain Z11[x]
6) How difficult was this homework? How long did it take?
2022-02-08