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The transcendence degree of Q(pi), sometimes called the transcendental degree, is one because it is generated by one extra element. In contrast, Q(pi,pi^2) (which is the same ...

The Kronecker-Weber theorem, sometimes known as the Kronecker-Weber-Hilbert theorem, is one of the earliest known results in class field theory. In layman's terms, the ...

Disconnectivities are mathematical entities which stand in the way of a space being contractible (i.e., shrunk to a point, where the shrinking takes place inside the space ...

The integral closure of a commutative unit ring R in an extension ring S is the set of all elements of S which are integral over R. It is a subring of S containing R.

Given a commutative unit ring R and an extension ring S, an element s of S is called integral over R if it is one of the roots of a monic polynomial with coefficients in R.

The ring of integers of a number field K, denoted O_K, is the set of algebraic integers in K, which is a ring of dimension d over Z, where d is the extension degree of K over ...

Let K be a number field of extension degree d over Q. Then an order O of K is a subring of the ring of integers of K with d generators over Z, including 1. The ring of ...

Given algebraic numbers a_1, ..., a_n it is always possible to find a single algebraic number b such that each of a_1, ..., a_n can be expressed as a polynomial in b with ...

A field F in which any Pythagorean extension of F coincides with F.

Given a rectangle BCDE, draw EF=DE on an extension of BE. Bisect BF and call the midpoint G. Now draw a semicircle centered at G, and construct the extension of ED which ...