In mathematics, an algebraic model defines a set in terms of the operations that can be applied to it; their meaning is defined purely in terms of the equivalences that hold between different applications.

A set of entities, operations, and the equivalences that hold for them, is an algebra.

Some familiar equivalences:

  • (a+b)*c = (a+c)*(b+c)
  • a*1 = 1*a = a

A typical algebraic approach: the Peano axioms to define natural numbers.

Let M be a field extension of a field K. An element a in M is called algebraic over K if there exists a nonzero polynomial f(x) with coefficients in K such that f(a)=0. Otherwise a is called transcendental over K.

For example, sqrt(2) is algebraic over Q but pi is not.

We say that M is algebraic over K if every element of M is algebraic over K. Whenever M is finite-dimensional when considered as a vector space over K it is algebraic over K. To see this take a in M and think about the powers of a: 1,a, a2,a3,.... Since these are elements of a finite-dimensional vector space they are linearly dependent. This gives a nonzero polynomial over K with a as a root.

For example,Q(sqrt(2)) is algebraic over Q.

Al`ge*bra"ic (#), Al`ge*bra"ic*al (#), a.

Of or pertaining to algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings.

Algebraic curve, a curve such that the equation which expresses the relation between the coordinates of its points involves only the ordinary operations of algebra; -- opposed to a transcendental curve.


© Webster 1913.
