minimal polynomial (field theory)
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In field theory, a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the minimal polynomial of an element of an extension field of a field is, roughly speaking, the
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
of lowest degree having coefficients in the smaller 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. 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 K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' 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 More specifically, ''J''''α'' is the kernel of the
ring homomorphism In mathematics, 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 that preserves addition, multiplication and multiplicative identity ...
from ''F'' 'x''to ''E'' which sends polynomials ''g'' to their value ''g''(''α'') at the element ''α''. Because it is the kernel of a ring homomorphism, ''J''''α'' is an ideal of the polynomial ring ''F'' 'x'' it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of ''F'' (which is
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
if ''F'' 'x''is regarded as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over ''F''). 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 , i.e. if the latter is not the zero ideal, then is called an
algebraic element In mathematics, if is an associative algebra over , then an element of is an algebraic element over , or just algebraic over , if there exists some non-zero polynomial g(x) \in K /math> with coefficients in such that . Elements of that are no ...
over , and there exists a
monic polynomial In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
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 or morphism 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 the ...
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 K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' 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.


Properties

Throughout this section, let ''E''/''F'' be a field extension over ''F'' as above, let ''α'' ∈ ''E'' be an algebraic element over ''F'' and let ''J''''α'' be the ideal of polynomials vanishing on ''α''.


Uniqueness

The minimal polynomial ''f'' of ''α'' is unique. To prove this, suppose that ''f'' and ''g'' are monic polynomials in ''J''''α'' of minimal degree ''n'' > 0. We have that ''r'' := ''f''−''g'' ∈ ''J''''α'' (because the latter is closed under addition/subtraction) and that ''m'' := deg(''r'') < ''n'' (because the polynomials are monic of the same degree). If ''r'' is not zero, then ''r'' / ''c''''m'' (writing ''c''''m'' ∈ ''F'' for the non-zero coefficient of highest degree in ''r'') is a monic polynomial of degree ''m'' < ''n'' such that ''r'' / ''c''''m'' ∈ ''J''''α'' (because the latter is closed under multiplication/division by non-zero elements of ''F''), which contradicts our original assumption of minimality for ''n''. We conclude that 0 = ''r'' = ''f'' − ''g'', i.e. that ''f'' = ''g''.


Irreducibility

The minimal polynomial ''f'' of ''α'' is irreducible, i.e. it cannot be factorized as ''f'' = ''gh'' for two polynomials ''g'' and ''h'' of strictly lower degree. To prove this, first observe that any factorization ''f'' = ''gh'' implies that either ''g''(''α'') = 0 or ''h''(''α'') = 0, because ''f''(''α'') = 0 and ''F'' is a field (hence also an
integral domain In mathematics, 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 setting for studying divisibilit ...
). Choosing both ''g'' and ''h'' to be of degree strictly lower than ''f'' would then contradict the minimality requirement on ''f'', so ''f'' must be irreducible.


Minimal polynomial generates ''J''''α''

The minimal polynomial ''f'' of ''α'' generates the ideal ''J''''α'', i.e. every '' g'' in ''J''''α'' can be factorized as ''g=fh'' for some ''h' '' in ''F'' 'x'' To prove this, it suffices to observe that ''F'' 'x''is a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
, because ''F'' is a field: this means that every ideal ''I'' in ''F'' 'x'' ''J''''α'' amongst them, is generated by a single element ''f''. With the exception of the zero ideal ''I'' = , the generator ''f'' must be non-zero and it must be the unique polynomial of minimal degree, up to a factor in ''F'' (because the degree of ''fg'' is strictly larger than that of ''f'' whenever ''g'' is of degree greater than zero). In particular, there is a unique monic generator ''f'', and all generators must be irreducible. When ''I'' is chosen to be ''J''''α'', for ''α'' algebraic over ''F'', then the monic generator ''f'' is the minimal polynomial of ''α''.


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 as
f(x) = \prod_ (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(L/K) with G/N where N = \text(\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(√2)

If ''F'' = Q, ''E'' = R, ''α'' = √2, 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'' = √2 is ''a''(''x'') = ''x'' − √2.


Q(√''d'')

In general, for the quadratic extension given by a square-free d, computing the minimal polynomial of an element a + b\sqrt can be found using Galois theory. Then
\begin f(x) &= (x - (a + b\sqrt))(x - (a - b\sqrt)) \\ &= x^2 - 2ax + (a^2 - b^2d) \end
in particular, this implies 2a \in \mathbb and a^2 - b^2d \in \mathbb. This can be used to determine \mathcal_ 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 \alpha = \sqrt then the Galois action on \sqrt stabilizes \alpha. Hence the minimal polynomial can be found using the quotient group \text(\mathbb(\sqrt,\sqrt)/\mathbb)/\text(\mathbb(\sqrt)/\mathbb).


Roots of unity

The minimal polynomials in Q 'x''of
roots of unity In mathematics, a root of unity 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 number theory, the theory of group char ...
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 prim ...
s. The roots of the minimal polynomial of 2cos(2/n) are twice the real part of the primitive roots of unity.


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 de ...
*
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 ...
* Minimal polynomial (linear algebra)


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)