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Field extension

From AoPSWiki

If K and L are fields and K\subseteq L, then L/K is said to be a field extension. We sometimes say that L is a field extension of K.

If L/K is a field extension, then L may be thought of as a vector space over K. The dimension of this vector space is called the degree of the extension, and is denoted by [L:K].

Given three fields K\subseteq L\subseteq M, then, if the degrees of the extensions M/L, L/K and M/K, are finite, then are related by the tower law: [M:K] = [M:L]\cdot[L:M]

One common way to construct an extension of a given field K is to consider an irreducible polynomial g(x) in the polynomial ring K[x], and then to form the quotient ring K(\alpha) = K[x]/\langle g(x)\rangle. Since g(x) is irreducible, \langle g(x)\rangle is a maximal ideal and so K(\alpha) is actually a field. We can embed K into this field by a\mapsto [a], and so we can view K(\alpha) as an extension of K. Now if we define \alpha as [x], then we can show that in K(\alpha), g(\alpha) = 0, and every element of K(\alpha) can be expressed as a polynomial in \alpha. We can thus think of K(\alpha) as the field obtained by 'adding' a root of g(x) to K.

It can be shown that [K(\alpha):K] = \deg g.

As an example of this, we can now define the complex numbers, \mathbb{C} by \mathbb{C} = \mathbb{R}[i] = \mathbb{R}[x]/\langle x^2+1\rangle.

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