In
field theory, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the minimal polynomial of an element of a
field is, roughly speaking, the
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s,
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s,
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, or others.
More formally, a minimal polynomial is defined relative to a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
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
In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of which are not algebraic ove ...
over , and there exists a
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\ ...
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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
, where is the ideal of generated by . Minimal polynomials are also used to define
conjugate elements.
Definition
Let ''E''/''F'' be a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
, ''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
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
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 domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
s, 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
the minimal polynomial of any
not in
can be computed as
if
has no stabilizers in the Galois action. Since it is irreducible, which can be deduced by looking at the roots of
, it is the minimal polynomial. Note that the same kind of formula can be found by replacing
with
where
is the stabilizer group of
. For example, if
then its stabilizer is
, hence
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
, computing the minimal polynomial of an element
can be found using Galois theory. Then
in particular, this implies
and
. This can be used to determine
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
then the Galois action on
stabilizes
. Hence the minimal polynomial can be found using the quotient group
.
Roots of unity
The minimal polynomials in Q
'x''of
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
are the
cyclotomic polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitiv ...
s.
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
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
*
Algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
*
Minimal polynomials of
References
*
*
* Pinter, Charles C. ''A Book of Abstract Algebra''. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270–273. {{isbn, 978-0-486-47417-5
Polynomials
Field (mathematics)