with ai not necessarily distinct and such that the roots ai 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 as the Galois group of p (if we assume it is separable).
A splitting field of a set P of polynomials is the smallest field over which each of the polynomials in P splits.
An extension L that is a splitting field for a set of polynomials p(X) over K is called a normal extension of K. 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 immediate.
On the other hand, the existence of algebraic closures in general is often proved by 'passing to the limit' from the splitting field result, which therefore requires an independent proof 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.
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.
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
Since p(X) has at most n roots the construction will require at most n extensions.
The steps for constructing Ki are given as follows: The irreducible factor f(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 Ki [X] and Ki [X] / (f(X)) is, in fact, a field, the residue field for that maximal ideal.
be the natural projection of the ring onto its quotient then so π(X) is a root of f(X) and of p(X).
is equal to the degree of the irreducible factor f(X).
Its elements are of the form where the cj are in Ki and α = π(X).
(If one considers Ki +1 as a vector space over Ki then the powers α j for 0 ≤ j ≤ n−1 form a basis.)
The elements of Ki +1 can be considered as polynomials in α of degree less than n. Addition in Ki +1 is given by the rules for polynomial addition, and multiplication is given by polynomial multiplication modulo f(X).
First let 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 If the product g(α)h(α) has a term αm with m ≥ n it can be reduced as follows: As an example of the reduction rule, take Ki = Q[X], the ring of polynomials with rational coefficients, and take f(X) = X 7 − 2.
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.
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: If we identify a + bx with (a,b) then we see that addition and multiplication are given by 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 If we identify a + bi with (a, b) then we see that addition and multiplication are given by 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.
Such a quotient is a primitive cube root of unity—either
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 Note that applying the construction process outlined in the previous section to this example, one begins with