Motivation

Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, such as x² + 1 over R, the real numbers, have no roots. 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 chain of fields ${\displaystyle F=K_{0}\subset K_{1}\subset \cdots \subset K_{r-1}\subset K_{r}=K}$ 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 ${\displaystyle f_{1}(X)f_{2}(X)\cdots f_{k}(X)}$.
• 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(X) 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 of K_i[X] and K_i[X] / (f(X)) is, in fact, a field. Moreover, if we let ${\displaystyle \pi :K_{i}[X]\to K_{i}[X]/(f(X))}$ be the natural projection of the ring onto its quotient then

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

so π(X) is a root of f(X) and of p(X).

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

${\displaystyle c_{n-1}\alpha ^{n-1}+c_{n-2}\alpha ^{n-2}+\cdots +c_{1}\alpha +c_{0}}$

where the c_j are in K_i and α = π(X). (If one considers K_i +1 as a vector space over K_i then the powers α^j for 0 ≤ j ≤ n−1 form a basis.)

The elements of K_i +1 can be considered as polynomials in α 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(α) and h(α) in K_i +1 their product is g(α)h(α) = r(α) where r(X) is the remainder of g(X)h(X) when 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(α) = g(α)h(α) directly. First let

${\displaystyle f(X)=X^{n}+b_{n-1}X^{n-1}+\cdots +b_{1}X+b_{0}.}$

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

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

If the product g(α)h(α) has a term α^m with m ≥ n it can be reduced as follows:

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

As an example of the reduction rule, take K_i = Q[X], the ring of polynomials with rational coefficients, and take f(X) = X⁷ − 2. Let ${\displaystyle g(\alpha )=\alpha ^{5}+\alpha ^{2}}$ and h(α) = α³ +1 be two elements of Q[X]/(X⁷ − 2). The reduction rule given by f(X) is α⁷ = 2 so

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

The complex numbers

Consider the polynomial ring R[x], and the irreducible polynomial x² + 1. The quotient ring R[x] / (x² + 1) is given by the congruence x² ≡ −1. As a result, the elements (or equivalence classes) of R[x] / (x² + 1) are of the form a + bx where a and b belong to R. To see this, note that since x² ≡ −1 it follows that x³ ≡ −x, x⁴ ≡ 1, x⁵ ≡ x, etc.; and so, for example p + qx + rx² + sx³ ≡ p + qx + r(−1) + s(−x) = (p − r) + (q − s)x.

The addition and multiplication operations are given by firstly using ordinary polynomial addition and multiplication, but then reducing modulo x² + 1, i.e. using the fact that x² ≡ −1, x³ ≡ −x, x⁴ ≡ 1, x⁵ ≡ x, etc. Thus:

${\displaystyle (a_{1}+b_{1}x)+(a_{2}+b_{2}x)=(a_{1}+a_{2})+(b_{1}+b_{2})x,}$

${\displaystyle (a_{1}+b_{1}x)(a_{2}+b_{2}x)=a_{1}a_{2}+(a_{1}b_{2}+b_{1}a_{2})x+(b_{1}b_{2})x^{2}\equiv (a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})x\,.}$

If we identify a + bx with (a,b) then we see that addition and multiplication are given by

${\displaystyle (a_{1},b_{1})+(a_{2},b_{2})=(a_{1}+a_{2},b_{1}+b_{2}),}$

${\displaystyle (a_{1},b_{1})\cdot (a_{2},b_{2})=(a_{1}a_{2}-b_{1}b_{2},a_{1}b_{2}+b_{1}a_{2}).}$

We claim that, as a field, the quotient ring R[x] / (x² + 1) is isomorphic to the complex numbers, C. A general complex number is of the form a + bi, where a and b are real numbers and i² = −1. Addition and multiplication are given by

${\displaystyle (a_{1}+b_{1}i)+(a_{2}+b_{2}i)=(a_{1}+a_{2})+i(b_{1}+b_{2}),}$

${\displaystyle (a_{1}+b_{1}i)\cdot (a_{2}+b_{2}i)=(a_{1}a_{2}-b_{1}b_{2})+i(a_{1}b_{2}+a_{2}b_{1}).}$

If we identify a + bi with (a, b) then we see that addition and multiplication are given by

${\displaystyle (a_{1},b_{1})+(a_{2},b_{2})=(a_{1}+a_{2},b_{1}+b_{2}),}$

${\displaystyle (a_{1},b_{1})\cdot (a_{2},b_{2})=(a_{1}a_{2}-b_{1}b_{2},a_{1}b_{2}+b_{1}a_{2}).}$

The previous calculations show that addition and multiplication behave the same way in R[x] / (x² + 1) and C. In fact, we see that the map between R[x] / (x² + 1) and C given by a + bx → a + bi is a homomorphism with respect to addition and multiplication. It is also obvious that the map a + bx → a + bi is both injective and surjective; meaning that a + bx → a + bi is a bijective homomorphism, i.e., an isomorphism. It follows that, as claimed: R[x] / (x² + 1) ≅ C.

In 1847, Cauchy used this approach to define the complex numbers.^[1]

Cubic example

Let K be the rational number field Q and p(x) = x³ − 2. Each root of p equals ³√2 times a cube root of unity. Therefore, if we denote the cube roots of unity by

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

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

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

any field containing two distinct roots of p will contain the quotient between two distinct cube roots of unity. Such a quotient is a primitive cube root of unity—either ${\displaystyle \omega _{2}}$ or ${\displaystyle \omega _{3}=1/\omega _{2}}$. It follows that a splitting field L of p will contain ω₂, as well as the real cube root of 2; conversely, any extension of Q containing these elements contains all the roots of p. Thus

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

Note that applying the construction process outlined in the previous section to this example, one begins with ${\displaystyle K_{0}=\mathbf {Q} }$ and constructs the field ${\displaystyle K_{1}=\mathbf {Q} [X]/(X^{3}-2)}$. This field is not the splitting field, but contains one (any) root. However, the polynomial ${\displaystyle Y^{3}-2}$ is not irreducible over ${\displaystyle K_{1}}$ and in fact:

${\displaystyle Y^{3}-2=(Y-X)(Y^{2}+XY+X^{2}).}$

Note that ${\displaystyle X}$ is not an indeterminate, and is in fact an element of ${\displaystyle K_{1}}$. Now, continuing the process, we obtain ${\displaystyle K_{2}=K_{1}[Y]/(Y^{2}+XY+X^{2})}$ which is indeed the splitting field and is spanned by the ${\displaystyle \mathbf {Q} }$-basis ${\displaystyle \{1,X,X^{2},Y,XY,X^{2}Y\}}$. Notice that if we compare this with ${\displaystyle L}$ from above we can identify ${\displaystyle X={\sqrt[{3}]{2}}}$ and ${\displaystyle Y=\omega _{2}}$.

Other examples

• The splitting field of x^q − x over F_p is the unique finite field F_q for q = pⁿ.^[2] Sometimes this field is denoted by GF(q).

• The splitting field of x² + 1 over F₇ is F₄₉; the polynomial has no roots in F₇, i.e., −1 is not a square there, because 7 is not congruent to 1 modulo 4.^[3]

• The splitting field of x² − 1 over F₇ is F₇ since x² − 1 = (x + 1)(x − 1) already splits into linear factors.

• We calculate the splitting field of f(x) = x³ + x + 1 over F₂. It is easy to verify that f(x) has no roots in F₂, hence f(x) is irreducible in F₂[x]. Put r = x + (f(x)) in F₂[x]/(f(x)) so F₂(r ) is a field and x³ + x + 1 = (x + r)(x² + ax + b) in F₂(r )[x]. Note that we can write + for − since the characteristic is two. Comparing coefficients shows that a = r and b = 1 + r². The elements of F₂(r ) can be listed as c + dr + er², where c, d, e are in F₂. There are eight elements: 0, 1, r, 1 + r, r², 1 + r², r + r² and 1 + r + r². Substituting these in x² + rx + 1 + r² we reach (r²)² + r(r²) + 1 + r² = r⁴ + r³ + 1 + r² = 0, therefore x³ + x + 1 = (x + r)(x + r²)(x + (r + r²)) for r in F₂[x]/(f(x)); E = F₂(r ) is a splitting field of x³ + x + 1 over F₂.