# Difference between revisions of "Field extension"

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

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

It can be shown that <math>[K(\alpha):K] = \deg g</math>. | It can be shown that <math>[K(\alpha):K] = \deg g</math>. | ||

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

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## Latest revision as of 23:46, 24 March 2009

If and are fields and , then is said to be a **field extension**. We sometimes say that is a field extension of .

If is a field extension, then may be thought of as a vector space over . The dimension of this vector space is called the *degree* of the extension, and is denoted by .

Given three fields , then, if the degrees of the extensions , and , are finite, then are related by the tower law:

One common way to construct an extension of a given field is to consider an irreducible polynomial in the polynomial ring , and then to form the quotient ring . Since is irreducible, is a maximal ideal and so is actually a field. We can embed into this field by , and so we can view as an extension of . Now if we define as , then we can show that in , , and every element of can be expressed as a polynomial in . We can thus think of as the field obtained by 'adding' a root of to .

It can be shown that .

As an example of this, we can now *define* the complex numbers, by .

*This article is a stub. Help us out by expanding it.*