Conjugate (algebra)

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This article is about conjugates in field extensions. For other uses, see conjugate.

In algebra, a conjugate of an element in a quadratic extension field of a field K is its image under the unique non-identity automorphism of the extended field that fixes K. If the extension is generated by a square root of an element r of K, then the conjugate of $a+b\sqrt r$ is $a-b\sqrt r$ for $a,b\in K$ , and in particular in the case of the field C of complex numbers as an extension of the field R of real numbers (where $r = - 1$ ), the complex conjugate of $a + b i$ is $a - b i$ .

Forming the sum or product of any element of the extension field with its conjugate always gives an element of K. This can be used to rewrite a quotient of numbers in the extended field so that the denominator lies in K, by multiplying numerator and denominator by the conjugate of the denominator. This process is called rationalization of the denominator, in particular if K is the field Q of rational numbers.

[edit] Differences of squares

An expression of the form

$a^2-b^2 \,$

can be factored to give

$(a+b)(a-b) \,$

where one factor is the conjugate of the other.

This can be useful when trying to rationalize a denominator containing a radical.

[edit] Rationalizing radicals in the denominator

An irrational denominator of the form $a+b\sqrt r$ can be made rational by multiplying numerator and denominator by the conjugate $a-b\sqrt r$ , so that the denominator becomes $a^2+ab\sqrt r-ab\sqrt r-b^2r$ , or $a 2 - b 2 r$ .

$\left ( \frac{1}{a+b\sqrt r} \right ) \left ( \frac{a-b\sqrt r}{a-b\sqrt r} \right ) =\frac{a-b\sqrt r}{a^2-b^2r}$

Here is an example:

$\left ( \frac{1}{2+2\sqrt 3} \right ) \left ( \frac{2-2\sqrt 3}{2-2\sqrt 3} \right ) =\frac{2-2\sqrt 3}{2^2-2^2\cdot 3}=\frac{2\sqrt 3-2}{8}=\frac{\sqrt 3-1}{4} \,$