An algebraic element over a field in an extension field is an element that is a root of some nonzero polynomial with coefficients in .

Every element of is algebraic over . The sum, difference, product, and quotient of algebraic elements in are again algebraic over , with the denominator of a quotient required to be nonzero. It follows that the simple extension field is an algebraic extension of iff is algebraic over .

The imaginary unit i is algebraic over the field of real numbers since it is a root of the polynomial . Because its coefficients are integers, it is even true that is algebraic over the field of rational numbers, i.e., it is an algebraic number (and also an algebraic integer). As a consequence, and are algebraic extensions of and respectively. (Here, is the complex field , whereas is the total ring of fractions of the ring of Gaussian integers .)

An algebraic simple extension field is always a finite extension. Its extension field degree is the polynomial degree of the extension field minimal polynomial of over . More generally, any finite list of algebraic elements generates a finite extension, which explains closure under the arithmetic operations above (Milne 2022, pp. 18-19).

See also

Integral Element, Transcendental Element

Portions of this entry contributed by Margherita Barile

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References

Milne, J. S. Fields and Galois Theory. Version 5.10, Sep. 2022. https://www.jmilne.org/math/CourseNotes/FT.pdf.

Referenced on Wolfram|Alpha

Algebraic Element

Cite this as:

Weisstein, Eric W., with contributions by Margherita Barile. "Algebraic Element." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraicElement.html

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