Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors (or "splits", hence the name) into linear factors.

Definition

A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors into linear factors

$p(X)=\prod _{i=1}^{\deg(p)}(X-a_{i})\in L[X]$

and such that the coefficients a_i generate L over K. The extension L is then an extension of minimal degree over K in which p splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known to be the Galois group of p (if we assume it is separable).

Facts

An extension L which is a splitting field for multiple polynomials p(X) over K is called a normal extension.

Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is automatic. On the other hand, the existence of algebraic closures in general is usually proved by 'passing to the limit' from the splitting field result; which is therefore proved directly to avoid circular reasoning.

Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for all the polynomials p over K that are minimal polynomials over K of elements a of K′.

Constructing splitting fields

Motivation

Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, have no roots such as $X^{2}+1$ over $\mathbb {R}$, the real numbers. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.

The Construction

Let F be a field and p(X) be a polynomial in the polynomial ring F[X] of degree n. The general process for constructing K, the splitting field of p(X) over F, is to construct a sequence of fields $F=K_{0},K_{1},\ldots K_{r-1},K_{r}=K$ such that $K_{i}$ is an extension of $K_{i-1}$ containing a new root of p(X). Since p(X) has at most n roots the construction will require at most n extensions. The steps for constructing $K_{i}$ are given as follows:

• Factorize p(X) over $K_{i}$ into irreducible factors $f_{1}(X)f_{2}(X)\cdots f_{k}(X)$.
• Choose any nonlinear irreducible factor $f(X)=f_{i}(X)$.
• Construct the field extension $K_{i+1}$ of $K_{i}$ as the quotient ring $K_{i+1}=K_{i}[X]/(f(X))$ where (f(X)) denotes the ideal in $K_{i}[X]$ generated by f(X)
• Repeat the process for $K_{i+1}$ until p(X) completely factors.

The irreducible factor $f_{i}$ used in the quotient construction may be chosen arbitrarily. Although different choices of factors may lead to different subfield sequences the resulting splitting fields will be isomorphic.

Since f(X) is irreducible, (f(X)) is a maximal ideal and hence $K_{i}[X]/(f(X))$ is, in fact, a field. Moreover, if we let $\pi :K_{i}[X]\to K_{i}[X]/(f_{1}(X))$ be the natural projection of the ring onto its quotient then

$f(\pi (X))=\pi (f(X))=f(X)\ {\bmod {\ }}f(X)=0$

so $\pi (X)$ is a root of f(X) and of p(X).

The degree of a single extension $[K_{i+1}:K_{i}]$ is equal to the degree of the irreducible factor f(X). The degree of the extension [K : F] is given by $[K_{r}:K_{r-1}]\cdots [K_{2}:K_{1}][K_{1}:F]$ and is at most n!.

The Field $K_{i}[x]/(f(x))$

As mentioned above, the quotient ring $K_{i+1}=K_{i}[X]/(f(X))$ is a field when f(X) is irreducible. Its elements are of the form $c_{n-1}\alpha ^{n-1}+c_{n-2}\alpha ^{n-2}+\cdots +c_{1}\alpha +c_{0}$ where the $c_{j}$ are in $K_{i}$ and $\alpha =\pi (X)$. (If one considers $K_{i+1}$ as a vector space over $K_{i}$ then the powers $\alpha ^{j}$ for 1 ≤ j ≤ n−1 form a basis.)

The elements of $K_{i+1}$ can be considered as polynomials in $\alpha$ of degree less than n. Addition in $K_{i+1}$ is given by the rules for polynomial addition and multiplication is given by polynomial multiplication modulo f(X). That is, for $g(\alpha )$ and $h(\alpha )$ in $K_{i+1}$ the product $g(\alpha )h(\alpha )=r(\alpha )$ where r(X) is the remainder of g(X)h(X) divided by f(X) in $K_{i}[X]$.

The remainder r(X) can be computed through long division of polynomials, however there is also a straightforward reduction rule that can be used to compute $r(\alpha )=g(\alpha )h(\alpha )$ directly. First let $f(X)=X^{n}+b_{n-1}X^{n-1}+\cdots +b_{1}X+b_{0}$. (The polynomial is over a field so one can take f(X) to be monic without loss of generality.) Now α is a root of f(X), so $\alpha ^{n}=-(b_{n-1}\alpha ^{n-1}+\cdots +b_{1}\alpha +b_{0})$. If the product $g(\alpha )h(\alpha )$ has a term $\alpha ^{m}$ with m ≥ n it can be reduced as follows:

$\alpha ^{n}\alpha ^{m-n}=-(b_{n-1}\alpha ^{n-1}+\cdots +b_{1}\alpha +b_{0})\alpha ^{m-n}=-(b_{n-1}\alpha ^{m-1}+\cdots +b_{1}\alpha ^{m-n+1}+b_{0}\alpha ^{m-n+1})$.

As an example of the reduction rule, take $K_{i}=\mathbb {Q} [X]$, the ring of polynomials with rational coefficients, and take $f(X)=X^{7}-2$. Let $g(\alpha )=\alpha ^{5}+\alpha ^{2}$ and $h(\alpha )=\alpha ^{3}+1$ be two elements of $\mathbb {Q} [X]/(X^{7}-2)$. The reduction rule given by f(X) is $\alpha ^{7}=2$ so

$g(\alpha )h(\alpha )=\left(\alpha ^{5}+\alpha ^{2}\right)\left(\alpha ^{3}+1\right)=\alpha ^{8}+2\alpha ^{5}+\alpha ^{2}=\left(\alpha ^{7}\right)\alpha +2\alpha ^{5}+\alpha ^{2}=2\alpha ^{5}+\alpha ^{2}+2\alpha .$

Examples

If K is the rational number field Q and

p(X) = X³ − 2,

then a splitting field L will contain a primitive cube root of unity, as well as a cube root of 2. Thus

${L=\mathbb {Q} ({\sqrt[{3}]{2}},\omega _{2})=\{a+b\omega _{2}+c{\sqrt[{3}]{2}}+d{\sqrt[{3}]{2}}\omega _{2}+e{\sqrt[{3}]{2^{2}}}+f{\sqrt[{3}]{2^{2}}}\omega _{2}\,|\,a,b,c,d,e,f\in \mathbb {Q} \}}$

where

$\omega _{1}=1\,$,

$\omega _{2}=-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i$, and

$\omega _{3}=-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i$

are the cubic roots of unity.

Constructing a field isomorphic to the complexes

Let $f=X^{2}+1\in \mathbb {R} [X]$. Then $\mathbb {R} [X]/(f)$, where X is the equivalence class of X modulo f, has:

• elements: $a+bX,\quad a,b\in \mathbb {R}$;
• addition: $(a_{1}+b_{1}X)+(a_{2}+b_{2}X)=(a_{1}+a_{2})+(b_{1}+b_{2})X$;
• multiplication: $(a_{1}+b_{1}X)(a_{2}+b_{2}X)=(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+a_{2}b_{1})X$.

Exchanging i for X recovers the field $\mathbb {C}$ of complex numbers, which shows that $\mathbb {R} [X]/(f)$ is isomorphic to $\mathbb {C}$.

Other examples

• A splitting field of $x^{2}+1$ over $\mathbb {F} _{7}$ is $\mathbb {F} _{49}$; as 7 is not equivalent to 1 (mod 4), −1 has no square root over $\mathbb {F} _{7}$.
• The splitting field of $x^{2}-1$ over $\mathbb {F} _{7}$ is $\mathbb {F} _{7}$ since $x^{2}-1=(x+1)(x-1)$ already factors into linear factors.

See also

• Rupture field

References

• Dummit, David S., and Foote, Richard M. (1999). Abstract Algebra (2nd ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-36857-1.
• Weisstein, Eric W. "Splitting field". MathWorld.