- 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, E, and F[x] a minimal polynomial. Suppose f = g * h where 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 the minimal polynomial for α is p(x) = x2 − 2.
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 , then
is the minimal polynomial for
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