Minimal polynomial (field theory)


Minimal polynomial (field theory)

In field theory, given a field extension E / F and an element α of E that is an algebraic element 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.

Proof: Let E / F be a field extension over F as above, \alpha \in E, and f \in F[x] a minimal polynomial. Suppose f = g * h where g,h \in F[x]\F. Hence f(α) = 0. Since fields are also integral domains, we have that g(α) = 0 or h(α) = 0. As both the degrees of both g and h are smaller than the degree of f, we get a contradiction as f does not have a minimal degree. We conclude that minimal polynomials are irreducible.

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.

References


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