In field theory, a branch of mathematics, the minimal polynomial of an element  of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that  is a root of the polynomial.
If the minimal polynomial of  exists, it is unique.
The coefficient of the highest-degree term in the polynomial is required to be 1, and the 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  and an element of the extension field .
The minimal polynomial of an element, if it exists, is a member of , the  ring of polynomials in the variable  with coefficients in .
Given an element  of , let  be the set of all polynomials  in  such that .
The element  is called a root or zero of each polynomial in .
The set  is so named because it is an ideal of .
The zero polynomial, all of whose coefficients are 0, is in every  since  for all  and .
This makes the zero polynomial useless for classifying different values of  into types, so it is excepted.
If there are any non-zero polynomials in , then  is called an algebraic element over , and there exists a monic polynomial of least degree in .
This is the minimal polynomial of  with respect to .
It is unique and irreducible over .
If the zero polynomial is the only member of , then  is called a transcendental element over  and has no minimal polynomial with respect to .
Minimal polynomials are useful for constructing and analyzing field extensions.
When  is algebraic with minimal polynomial , the smallest field that contains both  and  is isomorphic to the quotient ring , where  is the ideal of  generated by .
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 subα from F[x] to E that substitutes α for x, that is, subα(f(x)) = f(α).
The kernel of subα, ker(subα), is the set of all polynomials in F[x] that have α as a root.
That is, ker(subα) = Jα from above.
Since subα is a ring homomorphism, ker(subα) 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(subα) that generates ker(subα).
Such a polynomial will have least degree among all non-zero polynomials in ker(subα), 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 Jα of minimal degree n > 0.
Since p − q ∈ Jα 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 L/K the minimal polynomial of any \alpha \in L not in K can be computed asf(x) = \prod_{\sigma \in \text{Gal}(L/K)} (x - \sigma(\alpha))if \alpha has no stabilizers in the Galois action.
Since it is irreducible, which can be deduced by looking at the roots of f', it is the minimal polynomial.
Note that the same kind of formula can be found by replacing G = \text{Gal}(L/K) with G/N where N = \text{Stab}(\alpha) is the stabilizer group of \alpha.
For example, if \alpha \in K then its stabilizer is G, hence (x-\alpha) is its minimal polynomial.
Quadratic field extensions
Q({{radic|2}})
If F = Q, E = R, α = , then the minimal polynomial for α is a(x) = x2 − 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({{radic|d}})
In general, for the quadratic extension given by a square-free d, computing the minimal polynomial of an element a + b\sqrt{d} can be found using Galois theory.
Then\begin{align} f(x) &= (x - (a+b\sqrt{d}))(x - (a - b\sqrt{d})) \\ &= x^2 - 2ax + (a^2 - b^2d) \end{align}in particular, this implies 2a \in \mathbb{Z} and a^2 - b^2d \in \mathbb{Z}.
This can be used to determine \mathcal{O}_{\mathbb{Q}(\sqrt{d})} through a series of relations using modular arithmetic.
Biquadratic field extensions
If  α =  + , then the minimal polynomial in Q[x] is a(x) = x4 − 10x2 + 1 = (x −  −  )(x +  − )(x −  + )(x +  + ).
Notice if \alpha = \sqrt{2} then the Galois action on \sqrt{3} stabilizes \alpha.
Hence the minimal polynomial can be found using the quotient group \text{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q})/\text{Gal}(\mathbb{Q}(\sqrt{3})/\mathbb{Q}).
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.
See also
Ring of integers
Algebraic number field
Minimal polynomials of 2\cos(2\pi/n)
References
Pinter, Charles C.
A Book of Abstract Algebra.
Dover Books on Mathematics Series.
Dover Publications, 2010, p. 270–273.
