Algebra for Symbolic Computation by Antonio Machì

Author:Antonio Machì
Language: eng
Format: epub
Publisher: Springer Milan, Milano

Example.

Let f = x4 + 3x3 + 3x2 − 5. Modulo 2, this polynomial is x4 + x3 + x2 + 1, and admits the root 1. Dividing by x − 1 we find

the two factors being irreducible, the latter is the factorisation of f over Z2. Hence, if f factors over Z, this cannot happen with an irreducible factor of degree 2. Indeed, either such a factor remains irreducible over Z2, but this would yield a different factorisation over Z2, or it decomposes into two linear factors, which would give yet another factorisation mod 2. The only possibility is that, over Z, f = (ax + b)q(x), so that f has a linear factor, and so a root, modulo p for all p. But for p = 3, f = x4 + 1, and f(0) = 1, f(1) = 2 and f(2) = 2, so there are no roots over Z3. It follows that f is irreducible over Z.

A sufficient condition for the irreducibility of a polynomial is given in the following theorem.

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