7

Quadratic fields

**7.1 Algebraic number fields**

Although we shall be concerned principally in this chapter only with quadratic fields, we shall nevertheless begin with a short discussion of the more general concept of an algebraic number field. The theory relating to such fields arose from attempts to solve Fermat’s last theorem and it is one of the most beautiful and profound in mathematics.

Let *α* be an algebraic number with degree *n* and let *P* be the minimal polynomial for *α* (see Section 6.5). By the conjugates of *α* we mean the zeros *α*_{1}, …, *α*_{n} of *P*. The algebraic number field *k* generated by *α* over the rationals is defined as the set of numbers ...

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