Algebraic Numbers, An Introduction


As the name suggests, algebraic number theory employs modern algebraic techniques to solve problems in number theory. We are therefore interested in integral extensions of Z (the integers), and field extensions of Q (the rationals). However, other extensions will prove useful as well.

A global field is a finite separable extension of Q or of Zp(t). We are extending the rationals, or the quotients of polynomials in t over the integers mod p.

A number field is a finite extension of Q. Since Q has characteristic 0, extensions of Q are automatically separable. Thus a number field is a global field. In contrast, Zp(t) supports inseparable extensions, such as the pth root of t. We want to avoid those; hence a global field is declared separable.

within the context of a global field, an algebraic number is algebraic over the base field, either Q or Zp(t). Of course every element in a finite extension is algebraic, so everything in a global field qualifies as an algebraic number.

An algebraic integer is integral over the base ring, Z or Zp[t]. For example, sqrt(2) is an algebraic integer, while 1/3 is not.