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§ 6.4 Splitting fields

 
Definition 6.5.1. Let K be a field and let f(x) = a₀ + a₁ x + · · · + a_nxⁿ be a polynomial in K[x] of degree n>0. An extension field F of K is called a splitting field for f(x) over K if there exist elements r₁, r₂, . . . , r_n in F such that

(i) f(x) = a_n (x-r₁) (x-r₂) · · · (x-r_n), and

(ii) F = K(r₁,r₂,...,r_n).

 
In the above situation we usually say that f(x) splits over the field F. The elements r₁, r₂, . . . , r_n are roots of f(x), and so F is obtained by adjoining to K a complete set of roots of f(x).

 
Theorem 6.4.2. Let f(x) be a polynomial in K[x] of degree n>0. Then there exists a splitting field F for f(x) over K, with [F:K] n!.

 
Lemma 6.4.3. Let : K -> L be an isomorphism of fields. Let F be an extension field of K such that F = K(u) for an algebraic element u in F. Let p(x) be the minimal polynomial of u over K. If v is any root of the image q(x) of p(x) under , and E=L(v), then there is a unique way to extend to an isomorphism : F -> E such that (u) = v and (a) = (a) for all a in K.

 
Theorem 6.4.5. Let f(x) be a polynomial over the field K. The splitting field of f(x) over K is unique up to isomorphism.

§ 6.4 Solved problems

 
1. Find the splitting field over Q for the polynomial   x⁴ + 4. Solution

2. Find the degree of the splitting field over Z₂ for the polynomial   (x³ + x + 1)(x² + x + 1). Solution

3. Find the degree [F:Q], where F is the splitting field of the polynomial   x³ - 11   over the field Q of rational numbers. Solution

4. Determine the splitting field over Q for   x⁴ + 2. Solution

5. Determine the splitting field over Q for   x⁴ + x² + 1. Solution

6. Factor   x⁶ - 1   over Z₇;   factor   x⁵ - 1   over Z₁₁. Solution

Solutions to the problems | Forward to §6.5 | Back to §6.3 | Up | Table of Contents