Algebraic element

Template:Short description Template:No footnotes In mathematics, if Template:Math is an associative algebra over Template:Math, then an element Template:Math of Template:Math is an algebraic element over Template:Math, or just algebraic over Template:Math, if there exists some non-zero polynomial $g(x)\in K[x]$ with coefficients in Template:Math such that Template:Math. Elements of Template:Math that are not algebraic over Template:Math are transcendental over Template:Math. A special case of an associative algebra over $K$ is an extension field $L$ of $K$.

These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is Template:Math, with Template:Math being the field of complex numbers and Template:Math being the field of rational numbers).

Examples

• The square root of 2 is algebraic over Template:Math, since it is the root of the polynomial Template:Math whose coefficients are rational.
• Pi is transcendental over Template:Math but algebraic over the field of real numbers Template:Math: it is the root of Template:Math, whose coefficients (1 and −Template:Pi) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses Template:Math, not Template:Math.)

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 ${\epsilon }_a:K[X]\to K(a),P\mapsto P(a)$. 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 $K[X]/(p)\to \mathrm{i}\mathrm{m}({\epsilon }_a)$ obtained by the homomorphism theorem is an isomorphism of fields, where we can then observe that $\mathrm{i}\mathrm{m}({\epsilon }_a)=K(a)$. Otherwise, ${\epsilon }_a$ is injective and hence we obtain a field isomorphism $K(X)\to K(a)$, 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.

See also

• Algebraic independence

References

• Template:Lang Algebra