# 数学Homework代做 MAT401 Homework 4 ## MAT401Homework 4

### Read: Gallian, Chapters 17 and 18 (Omit the material on Euclidean domains) Problems:  数学Homework代做

These problems are due on Crowdmark by 6pm on Wednesday, February 16. The solutions will be discussed in tutorials that week.

1. Consider polynomials in R = Z2[x] and recall that non-constant polynomials in this ring have a unique factorization into irreducibles. In homework 3 you found the irreducibles of degree at most 3.
(i) Find all irreducible polynomials of degree 4. Explain your answer. (There are 3 of them.)
(ii) Find the factorization into irreducibles of the following polynomials:  数学Homework代做 1. Determine whether or not each of the following polynomials is irreducible over Q. Explain what method you used.  数学Homework代做 1. Suppose that r is a non-zero real number such that the quantity r + 1/r is three times an integer. Show that r is irrational. (i) Show that (ii) Show that the elements 2 and 1 + √5 are both irreducible in R. 数学Homework代做
(iii) Show that R is not a UFD. 