Investiga tion 37

Finite Fields, the Group of Units in

Z

n

,

and Splitting Fields

Focus Questions

By the end of this investigation, you should be able to give precise and thorough

answers to the questions listed below. You may want to keep these questions in mind

to focus your tho ughts as you complete the investigation.

• What is a ﬁnite ﬁeld? How many elements ca n a ﬁnite ﬁeld have?

• How are any two ﬁnite ﬁelds of the same order related?

• What is the structure o f the group U

n

of units in Z

n

?

• What is the splitting ﬁeld of a polynomial? Why do we refer to “th e” splitting ﬁeld

instead of “a” splitting ﬁeld?

• How can one construct the splitting ﬁeld of a polynomial?

Preview Activity 37.1. Let p(x) = x

2

+ x + [1] in Z

2

[x].

(a) Find all of the elements of K = Z

2

[x]/p(x).

(b) Cre a te the addition and multiplication tables for K.

(c) What special kind of rin g is K? Explain.

(d) Wh a t kind of gr oup is U (K), the group of units in K? Explain.

(e) Explain why K contains all of the roots of the polynomial f(x) = x

4

− x over Z

2

. In fact,

show that every element of K is a root of f(x) = x

4

− x over Z

2

.

Introduction

The structure of the gr oups U

n

, which consist of the units in Z

n

, is not obvious. However, with a bit

of work we can determine the struc ture of th ese groups. To do so, we will ﬁrst show that the group

U

p

is cyclic when p is prime. We will then use this result to describe every group U

n

, whether n

505

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