In field theory, a branch of mathematics, the minimal polynomial of an element α {\displaystyle \alpha } of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that α {\displaystyle \alpha } is a root of the polynomial. If the minimal polynomial of α {\displaystyle \alpha } exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1. More formally, a minimal polynomial is defined relative to a field extension E / F {\displaystyle E/F} and an element of the extension field E / F {\displaystyle E/F} . The minimal polynomial of an element, if it exists, is a member of F [ x ] {\displaystyle F[x]} , the ring of polynomials in the variable x {\displaystyle x} with coefficients in F {\displaystyle F} . Given an element α {\displaystyle \alpha } of E {\displaystyle E} , let J α {\displaystyle J_{\alpha }} be the set of all polynomials f ( x ) {\displaystyle f(x)} in F [ x ] {\displaystyle F[x]} such that f ( α ) = 0 {\displaystyle f(\alpha )=0} . The element α {\displaystyle \alpha } is called a root or zero of each polynomial in J α {\displaystyle J_{\alpha }} . More specifically, J α {\displaystyle J_{\alpha }} is the kernel of the ring homomorphism from F [ x ] {\displaystyle F[x]} to E {\displaystyle E} which sends polynomials g {\displaystyle g} to their value g ( α ) {\displaystyle g(\alpha )} at the element α {\displaystyle \alpha } . Because it is the kernel of a ring homomorphism, J α {\displaystyle J_{\alpha }} is an ideal of the polynomial ring F [ x ] {\displaystyle F[x]} : it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of F {\displaystyle F} (which is scalar multiplication if F [ x ] {\displaystyle F[x]} is regarded as a vector space over F {\displaystyle F} ). The zero polynomial, all of whose coefficients are 0, is in every J α {\displaystyle J_{\alpha }} since 0 α i = 0 {\displaystyle 0\alpha ^{i}=0} for all α {\displaystyle \alpha } and i {\displaystyle i} . This makes the zero polynomial useless for classifying different values of α {\displaystyle \alpha } into types, so it is excepted. If there are any non-zero polynomials in J α {\displaystyle J_{\alpha }} , i.e. if the latter is not the zero ideal, then α {\displaystyle \alpha } is called an algebraic element over F {\displaystyle F} , and there exists a monic polynomial of least degree in J α {\displaystyle J_{\alpha }} . This is the minimal polynomial of α {\displaystyle \alpha } with respect to E / F {\displaystyle E/F} . It is unique and irreducible over F {\displaystyle F} . If the zero polynomial is the only member of J α {\displaystyle J_{\alpha }} , then α {\displaystyle \alpha } is called a transcendental element over F {\displaystyle F} and has no minimal polynomial with respect to E / F {\displaystyle E/F} . Minimal polynomials are useful for constructing and analyzing field extensions. When α {\displaystyle \alpha } is algebraic with minimal polynomial f ( x ) {\displaystyle f(x)} , the smallest field that contains both F {\displaystyle F} and α {\displaystyle \alpha } is isomorphic to the quotient ring F [ x ] / ⟨ f ( x ) ⟩ {\displaystyle F[x]/\langle f(x)\rangle } , where ⟨ f ( x ) ⟩ {\displaystyle \langle f(x)\rangle } is the ideal of F [ x ] {\displaystyle F[x]} generated by f ( x ) {\displaystyle f(x)} . Minimal polynomials are also used to define conjugate elements.
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