## 305 -Homework set 3

This set is due February 25 at the beginning of lecture. Consult the syllabus for details on the homework policy.

1. Show directly that there is no field of $6$ elements. (“Directly” means, among other things, that you cannot use the facts mentioned without proof at the end of lecture 4.3.)

2. Construct a field of size $8.$ Once you are done, verify that all its elements satisfy the equation $x^8-x=0.$

3. Solve exercises 36–38 from Chapter 4 of the book.

4. Is the set $\{a+b\root 4\of 2 : a,b\in{\mathbb Q}\}$ a field with the usual $+,\times,0,1$?