## 305 -Homework set 3

February 18, 2009

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 $+,\times,0,1$?

## 305 -Fields (3)

February 18, 2009

At the end of last lecture we arrived at the question of whether every finite field is a ${\mathbb Z}_p$ for some prime $p.$

In this lecture we show that this is not the case, by exhibiting a field of 4 elements. We also find some general properties of finite fields. Finite fields have many interesting applications (in cryptography, for example), but we will not deal much with them as our focus through the course is on number fields, that we will begin discussing next lecture.

We begin by proving the following result:

Lemma 13. Suppose that ${\mathbb F}$ is a finite field. Then there is some natural number $n>0$ such that the sum of $n$ ones vanishes, $1+\dots+1=0.$ The least such $n$ is a prime that divides the size of the field