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Definition 8.5.1.
Let n be a positive integer, and let
be the complex number
= cos
+ i sin
, where
= 2
/ n.
The polynomial
_{n} (x) = _{k} (x - ^{k}),
where k belongs to the set of positive integers less than n and relatively prime to n, is called the nth cyclotomic polynomial.
Proposition 8.5.2.
Let n be a positive integer, and let
_{n}(x)
be the nth cyclotomic polynomial.
The following conditions hold:
Theorem 8.5.3.
The nth cyclotomic polynomial
_{n}(x)
is irreducible over Q,
for every positive integer n.
Theorem 8.5.4.
For every positive integer n,
the Galois group of the nth cyclotomic polynomial
_{n}(x)
over Q is isomorphic to
Z_{n}^{×}.
Theorem 8.5.6. [Wedderburn]
Any finite division ring is a field.
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