Abstract Algebra Theory and Applications - Computer Science ...

328 CHAPTER 19 FIELDSfor b i and c i in F . Theng(x) = (b 0 − c 0 ) + (b 1 − c 1 )x + · · · + (b n−1 − c n−1 )x n−1is in F [x] and g(α) = 0. Since the degree of g(x) is less than the degreeof p(x), the irreducible polynomial of α, g(x) must be the zero polynomial.Consequently,b 0 − c 0 = b 1 − c 1 = · · · = b n−1 − c n−1 = 0,or b i = c i for i = 0, 1, . . . , n − 1. Therefore, we have shown uniqueness.Example 7. Since x 2 + 1 is irreducible over R, 〈x 2 + 1〉 is a maximal idealin R[x]. So E = R[x]/〈x 2 +1〉 is a field extension of R that contains a root ofx 2 + 1. Let α = x + 〈x 2 + 1〉. We can identify E with the complex numbers.By Theorem 19.4, E is isomorphic to R(α) = {a + bα : a, b ∈ R}. We knowthat α 2 = −1 in E, sinceα 2 + 1 = (x + 〈x 2 + 1〉) 2 + (1 + 〈x 2 + 1〉)= (x 2 + 1) + 〈x 2 + 1〉= 0.Hence, we have an isomorphism of R(α) with C defined by the map thattakes a + bα to a + bi.Let E be a field extension of a field F . If we regard E as a vector spaceover F , then we can bring the machinery of linear algebra to bear on theproblems that we will encounter in our study of fields. The elements in thefield E are vectors; the elements in the field F are scalars. We can thinkof addition in E as adding vectors. When we multiply an element in Eby an element of F , we are multiplying a vector by a scalar. This view offield extensions is especially fruitful if a field extension E of F is a finitedimensional vector space over F , and Theorem 19.5 states that E = F (α)is finite dimensional vector space over F with basis {1, α, α 2 , . . . , α n−1 }.If an extension field E of a field F is a finite dimensional vector spaceover F of dimension n, then we say that E is a finite extension of degreen over F . We write[E : F ] = n.to indicate the dimension of E over F .Theorem 19.6 Every finite extension field E of a field F is an algebraicextension.□