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 00:46, 25 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
.
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