Algebraic element

In mathematics, if $A$ is an associative algebra over $K$ , then an element $a$ of $A$ is an algebraic element over $K$ , or just algebraic over $K$ , if there exists some non-zero polynomial $g(x)\in K[x]$ with coefficients in $K$ such that $g (a) = 0$ .^[1] Elements of $A$ that are not algebraic over $K$ are transcendental over $K$ . A special case of an associative algebra over $K$ is an extension field $L$ of $K$ . If all elements of $L$ are algebraic over $K$ , then $L/K$ is called an algebraic extension.

These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is $C / Q$ , with $C$ being the field of complex numbers and $Q$ being the field of rational numbers).

Examples

The square root of 2 is algebraic over $Q$ , since it is the root of the polynomial $g (x) = x 2 - 2$ whose coefficients are rational.
Pi is transcendental over $Q$ but algebraic over the field of real numbers $R$ : it is the root of $g (x) = x - π$ , whose coefficients (1 and − $π$ ) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses $C / Q$ , not $C / R$ .)
An indeterminate x in a field K(x) of rational functions or K((x)) of formal Laurent series is defined to be transcendental over K.
Algebraic functions are the algebraic elements over a field of rational functions K(x₁, ..., x_m).

Properties

The following conditions are equivalent for an element $a$ of an extension field $L$ of $K$ :

$a$ is algebraic over $K$ ,
the field extension $K(a)/K$ is algebraic, i.e. every element of $K(a)$ is algebraic over $K$ (here $K(a)$ denotes the smallest subfield of $L$ containing $K$ and $a$ ),
the field extension $K(a)/K$ has finite degree, i.e. the dimension of $K(a)$ as a $K$ -vector space is finite,
$K[a]=K(a)$ , where $K[a]$ is the set of all elements of $L$ that can be written in the form $g(a)$ with a polynomial $g$ whose coefficients lie in $K$ .

To make this more explicit, consider the polynomial evaluation $\varepsilon _{a}:K[X]\rightarrow K(a),\,P\mapsto P(a)$ . This is a homomorphism and its kernel is $\{P\in K[X]\mid P(a)=0\}$ . If $a$ is algebraic, this ideal contains non-zero polynomials, but as $K[X]$ is a euclidean domain, it contains a unique polynomial $p$ with minimal degree and leading coefficient $1$ , which then also generates the ideal and must be irreducible. The polynomial $p$ is called the minimal polynomial of $a$ and it encodes many important properties of $a$ . Hence the ring isomorphism $K[X]/(p)\rightarrow \mathrm {im} (\varepsilon _{a})$ obtained by the homomorphism theorem is an isomorphism of fields, where we can then observe that $\mathrm {im} (\varepsilon _{a})=K(a)$ . Otherwise, $\varepsilon _{a}$ is injective and hence we obtain a field isomorphism $K(X)\rightarrow K(a)$ , where $K(X)$ is the field of fractions of $K[X]$ , i.e. the field of rational functions on $K$ , by the universal property of the field of fractions. We can conclude that in any case, we find an isomorphism $K(a)\cong K[X]/(p)$ or $K(a)\cong K(X)$ . Investigating this construction yields the desired results.

This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over $K$ are again algebraic over $K$ . For if $a$ and $b$ are both algebraic, then $(K(a))(b)$ is finite. As it contains the aforementioned combinations of $a$ and $b$ , adjoining one of them to $K$ also yields a finite extension, and therefore these elements are algebraic as well. Thus set of all elements of $L$ that are algebraic over $K$ is a field that sits in between $L$ and $K$ .

Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed. The field of complex numbers is an example. If $L$ is algebraically closed, then the field of algebraic elements of $L$ over $K$ is algebraically closed, which can again be directly shown using the characterisation of simple algebraic extensions above. An example for this is the field of algebraic numbers.

References

^ Roman, Steven (2008). "18". Advanced Linear Algebra. Graduate Texts in Mathematics. New York, NY: Springer New York Springer e-books. pp. 458–459. ISBN 978-0-387-72831-5.

Algebraic element

Examples

Properties

See also

References

Further reading