Minimal polynomial (field theory)

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, \alpha \in E, and f \in F[x] a minimal polynomial. Suppose f = g * h where g,h \in 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 F = \mathbb{Q}, E = \mathbb{R}, \alpha = \sqrt 2 the minimal polynomial for α is p(x) = x2 − 2.


\alpha = \sqrt 2 + \sqrt 3


p(x) = x^4 - 10 x^2 + 1 = (x - \sqrt 2 - \sqrt 3)(x + \sqrt 2 - \sqrt 3)(x - \sqrt 2 + \sqrt 3)(x + \sqrt 2 + \sqrt 3)

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 F = \mathbb{R}, then

p(x) = x - \sqrt 2

is the minimal polynomial for

\alpha = \sqrt 2.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Minimal polynomial (linear algebra) — For the minimal polynomial of an algebraic element of a field, see Minimal polynomial (field theory). In linear algebra, the minimal polynomial μA of an n by n matrix A over a field F is the monic polynomial P over F of least degree such that… …   Wikipedia

  • Conjugate element (field theory) — Conjugate elements redirects here. For conjugate group elements, see Conjugacy class. In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the (other) roots of the minimal polynomial pK …   Wikipedia

  • Field extension — In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties. For… …   Wikipedia

  • Polynomial — In mathematics, a polynomial (from Greek poly, many and medieval Latin binomium, binomial [1] [2] [3], the word has been introduced, in Latin, by Franciscus Vieta[4]) is an expression of finite length constructed from variables (also known as… …   Wikipedia

  • Field norm — In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. Contents 1 Formal definitions 2 Example 3 Further properties 4 See also …   Wikipedia

  • Field trace — In mathematics, the field trace is a linear mapping defined for certain field extensions. If L / K is a finite Galois extension, it is defined for α in L as the sum of all the conjugates: g (α)of α, for g in the Galois group G of L over K . It is …   Wikipedia

  • Field of definition — In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K , it may not be obvious… …   Wikipedia

  • Algebraic number field — In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector… …   Wikipedia

  • Finite field — In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and… …   Wikipedia

  • Primitive polynomial — In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF( p m ). In other words, a polynomial F(X) with coefficients in GF( p ) = Z/ p Z is a primitive… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.