Properties
The following conditions are equivalent for an element a of L:
- a is algebraic over K
- the field extension K(a)/K has finite degree, i.e. the dimension of K(a) as a K-vector space is finite. (Here K(a) denotes the smallest subfield of L containing K and a)
- K = K(a), where K 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.
This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K. The set of all elements of L which are algebraic over K is a field that sits in between L and K.
If a is algebraic over K, then there are many non-zero polynomials g(x) with coefficients in K such that g(a) = 0. However there is a single one with smallest degree and with leading coefficient 1. This is the minimal polynomial of a and it encodes many important properties of a.
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.
Read more about this topic: Algebraic Element
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)