prov and math set theory

Math 410 Spring 2024 – Homework 6 — Due Friday, Apr 26 [A] Some rings have the “left cancellation property,” and others don’t. Here’s the definition, for a ring R: Left Cancellation Property: ∀ a, b, c ∈ R with a 6= 0, if ab = ac then b = c. In a commutative ring, left cancellation is equivalent to right cancellation, and we just call it the cancellation property. Let R be a commutative ring. Prove that if R has the cancellation property, then R has no zero divisors. (Theorem 25.1 proves something like the converse of this implication. So it’s related, but not particularly helpful for this exercise.) [B] EME #46 Include a proof that your function in part (a) is well-defined. [C] EME #52 [D] textbook #35.10 (use the remainder theorem). [E] textbook #36.1 [F] textbook #36.15 [G] (a) In š3 [x], the only monic degree one polynomials are x + 1, x + 2, and x. Notice that these can also be written as x − 2, x − 1, and x − 0, respectively (since [1]3 = [−2]3 , etc.) Which monic degree 1 polynomials are factors of x4 + 2 in š3 [x]? (Use the Factor Theorem, bottom of p. 167). (b) In š3 [x], write x4 + 2 as a product of irreducible polynomials. (c) Explain why, for any field F, a degree 3 polynomial in F[x] with no degree 1 factors must be irreducible. (It isn’t the same for polynomials of degree 4. For example, in ’[x], x4 + 2x + 1 = (x2 + 1)2 is a reducible degree 4 polynomial with no real roots, and hence no degree 1 factors in ’[x].) (d) Show that in š5 [x], x3 + x + 1 is irreducible.