Vipul: Created page with '==Definition== Let $n$ be a natural number. The '''cyclotomic polynomial''' of degree $n$ , denoted $\Phi_n$ , is defined in the following way...'

2009-05-15T01:34:59Z

Created page with '==Definition== Let <math>n</math> be a natural number. The '''cyclotomic polynomial''' of degree <math>n</math>, denoted <math>\Phi_n</math>, is defined in the following way...'

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==Definition==

Let <math>n</math> be a [[natural number]]. The '''cyclotomic polynomial''' of degree <math>n</math>, denoted <math>\Phi_n</math>, is defined in the following ways:

* It is the product, over all primitive <math>n^{th}</math> roots <math>\zeta</math> of <math>1</math>, of the linear polynomials <math>x - \zeta</math>.
* It is the [[minimal polynomial]] of any primitive <math>n^{th}</math> root of unity.

===Equivalence of definitions===

The equivalence of definitions essentially follows by using definition (1) and showing, from that definition, that [[cyclotomic polynomials are irreducible]].

Cyclotomic polynomial - Revision history

Vipul: Created page with '==Definition== Let n be a natural number. The '''cyclotomic polynomial''' of degree n, denoted \Phi_n, is defined in the following way...'

Vipul: Created page with '==Definition== Let $n$ be a natural number. The '''cyclotomic polynomial''' of degree $n$ , denoted $\Phi_n$ , is defined in the following way...'