Properties
A minimal polynomial is irreducible. Let E/F be a field extension over F as above, α ∈ E, and f ∈ F a minimal polynomial for α. Suppose f = gh, where g,h ∈ F are of lower degree than f. Now f(α) = 0. Since fields are also integral domains, we have g(α) = 0 or h(α) = 0. This contradicts the minimality of the degree of f. Thus minimal polynomials are irreducible.
Read more about this topic: Minimal Polynomial (field Theory)
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