Algebraic Element

Given a field F and an extension field K superset= F, an element alpha in K is called algebraic over F if it is a root of some nonzero polynomial with coefficients in F.

Obviously, every element of F is algebraic over F. Moreover, the sum, difference, product, and quotient of algebraic elements are again algebraic. It follows that the simple extension field F(alpha) is an algebraic extension of F iff alpha is algebraic over F.

The imaginary unit i is algebraic over the field R of real numbers since it is a root of the polynomial x^2+1. Because its coefficients are integers, it is even true that i is algebraic over the field Q of rational numbers, i.e., it is an algebraic number (and also an algebraic integer). As a consequence, R(i) and Q(i) are algebraic extensions of R and Q respectively. (Here, R(i) is the complex field C, whereas Q(i) is the total ring of fractions of the ring of Gaussian integers Z[i].)

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This entry contributed by Margherita Barile

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

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