Minimal polynomial (field theory)

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For the minimal polynomial of a matrix, see Minimal polynomial (linear algebra).

In field theory, given a field extension E / F and an element α of E which is algebraic over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p(α) = 0. The minimal polynomial is irreducible over F, and any other non-zero polynomial f with f(α) = 0 is a (polynomial) multiple of p.

For example, for F = \mathbb{Q}, E = \mathbb{R}, \alpha = \sqrt 2 the minimal polynomial for α is p(x) = x2 − 2. If \alpha = \sqrt 2 + \sqrt 3 then

p(x) = x^4 - 10 x^2 + 1 = (x - \sqrt 2 - \sqrt 3)(x + \sqrt 2 - \sqrt 3)(x - \sqrt 2 + \sqrt 3)(x + \sqrt 2 + \sqrt 3)

is the minimal polynomial.

The base field F is important as it determines the possibilities for the coefficients of p(x). For instance if we take F = \mathbb{R}, then p(x) = x - \sqrt 2 is the minimal polynomial for \alpha = \sqrt 2.

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