Minimal polynomial (field theory)

In field theory, a branch of mathematics, the minimal polynomial of a value "α" is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that "α" is a root of the polynomial. If the minimal polynomial of "α" exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1, and the specified type for the remaining coefficients could be integers, rational numbers, real numbers, or others.
More formally, a minimal polynomial is defined relative to a field extension "E"/"F" and an element of the extension field "E". The minimal polynomial of an element, if it exists, is a member of "F"["x"], the ring of polynomials in the variable "x" with coefficients in "F". Given an element "α" of "E", let "J""α" be the set of all polynomials "f"("x") in "F"["x"] such that "f"("α") = 0. The element "α" is called a root or zero of each polynomial in "J""α". The set "J""α" is so named because it is an ideal of "F"["x"]. The zero polynomial, all of whose coefficients are 0, is in every "J""α" since 0"α""i" = 0 for all "α" and "i". This makes the zero polynomial useless for classifying different values of "α" into types, so it is excepted. If there are any non-zero polynomials in "J""α", then "α" is called an algebraic element over "F", and there exists a monic polynomial of least degree in "J""α". This is the minimal polynomial of "α" with respect to "E"/"F". It is unique and irreducible over "F". If the zero polynomial is the only member of "J""α", then "α" is called a transcendental element over "F" and has no minimal polynomial with respect to "E"/"F".
Minimal polynomials are useful for constructing and analyzing field extensions. When "α" is algebraic with minimal polynomial "a"("x"), the smallest field that contains both "F" and "α" is isomorphic to the quotient ring "F"["x"]/⟨"a"("x")⟩, where ⟨"a"("x")⟩ is the ideal of "F"["x"] generated by "a"("x"). Minimal polynomials are also used to define conjugate elements.
Definition.
Let "E"/"F" be a field extension, "α" an element of "E", and "F"["x"] the ring of polynomials in "x" over "F". The element "α" has a minimal polynomial when "α" is algebraic over "F", that is, when "f"("α") = 0 for some non-zero polynomial "f"("x") in "F"["x"]. Then the minimal polynomial of "α" is defined as the monic polynomial of least degree among all polynomials in "F"["x"] having "α" as a root.
Uniqueness.
Let "a"("x") be the minimal polynomial of "α" with respect to "E"/"F". The uniqueness of "a"("x") is established by considering the ring homomorphism sub"α" from "F"["x"] to "E" that substitutes "α" for "x", that is, sub"α"("f"("x")) = "f"("α"). The kernel of sub"α", ker(sub"α"), is the set of all polynomials in "F"["x"] that have "α" as a root. That is, ker(sub"α") = "J""α" from above. Since sub"α" is a ring homomorphism, ker(sub"α") is an ideal of "F"["x"]. Since "F"["x"] is a principal ring whenever "F" is a field, there is at least one polynomial in ker(sub"α") that generates ker(sub"α"). Such a polynomial will have least degree among all non-zero polynomials in ker(sub"α"), and "a"("x") is taken to be the unique monic polynomial among these.
Uniqueness of monic polynomial.
Suppose "p" and "q" are monic polynomials in "J""α" of minimal degree "n" > 0. Since "p" − "q" ∈ "J""α" and deg("p" − "q") < "n" it follows that "p" − "q" = 0, i.e. "p" = "q".
Properties.
A minimal polynomial is irreducible. Let "E"/"F" be a field extension over "F" as above, "α" ∈ "E", and "f" ∈ "F"["x"] a minimal polynomial for "α". Suppose "f" = "gh", where "g", "h" ∈ "F"["x"] 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.
Examples.
Minimal polynomial of a Galois field extension.
Given a Galois field extension formula_1 the minimal polynomial of any formula_2 not in formula_3 can be computed asformula_4if formula_5 has no stabilizers in the Galois action. Since it is irreducible, which can be deduced by looking at the roots of formula_6, it is the minimal polynomial. Note that the same kind of formula can be found by replacing formula_7 with formula_8 where formula_9 is the stabilizer group of formula_5. For example, if formula_11 then its stabilizer is formula_12, hence formula_13 is its minimal polynomial.
Quadratic field extensions.
Q().
If "F" = Q, "E" = R, "α" = , then the minimal polynomial for "α" is "a"("x") = "x"2 − 2. The base field "F" is important as it determines the possibilities for the coefficients of "a"("x"). For instance, if we take "F" = R, then the minimal polynomial for "α" = is "a"("x") = "x" − .
Q().
In general, for the quadratic extension given by a square-free formula_14, computing the minimal polynomial of an element formula_15 can be found using Galois theory. Thenformula_16in particular, this implies formula_17 and formula_18. This can be used to determine formula_19 through a series of relations using modular arithmetic. 
Biquadratic field extensions.
If "α" = + , then the minimal polynomial in Q["x"] is "a"("x") = "x"4 − 10"x"2 + 1 = ("x" − − )("x" + − )("x" − + )("x" + + ).
Notice if formula_20 then the Galois action on formula_21 stabilizes formula_5. Hence the minimal polynomial can be found using the quotient group formula_23.
Roots of unity.
The minimal polynomials in Q["x"] of roots of unity are the cyclotomic polynomials.
Swinnerton-Dyer polynomials.
The minimal polynomial in Q["x"] of the sum of the square roots of the first "n" prime numbers is constructed analogously, and is called a Swinnerton-Dyer polynomial.