Page 481 - Advanced Linear Algebra

An Introduction to Algebras 465

The elements of < that generate < , that is, the elements of order are called

the primitive th roots of unity. We denote the set of primitive th roots of

unity by + . Hence, if  + , then

+ ~¸ “ ² Á ³~ ¹
(
has size , where is the Euler phi function. The value ² ³ is defined to

² ³
be the number of positive integers less than or equal to and relatively prime to

.)
The th cyclotomic polynomial is defined by

8²%³ ~ ²% c $³

$+
Thus,
deg²8 ²%³³ ~ ² ³

Since every th root of unity is a primitive th root of unity for some “ and
since every primitive th root of unity for “ is also an th root of unity, we

deduce that
<~ +

“
where the union is a disjoint one. It follows that

%c ~ 8 ²%³

“
Finally, we show that 8²%³ is monic and has integer coefficients by induction

on . It is clear from the definition that 8 ² % ³ is monic. Since 8 ² % ³ ~ % c ,

the result is true for ~ . If is a prime, then all nonidentity th roots of unity

are primitive and so

%c
8 ² % ³ ~ ~ % c b % c2 b Ä b % b
%c
and the result holds for ~ . Assume the result holds for all proper divisors of
. Then

% c ~ 8 ²%³ 8 ²%³ ~ 8 ²%³9²%³

“

By the induction hypothesis, 9²%³ has integer coefficients and it follows that
8²%³ must also have integer coefficients.

Wedderburn's Theorem
Now we can prove Wedderburn's theorem.