# 代做数学Homework MAT401 Homework 4 ## MAT401 Homework 4

Read: Gallian, Chapters 17 and 18 (Omit the material on Euclidean domains)

### Problems:  代做数学Homework

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: 1. Determine whether or not each of the following polynomials is irreducible over Q. Explain what method you used.  代做数学Homework 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. (ii) Show that the elements 2 and 1 + 5 are both irreducible inR.

(iii) Show that R is not a UFD.  代做数学Homework 