**Number Fields**

Suppose *f* is an *k*-degree polynomial over **Q** (the rational numbers), and *r* is a complex root of *f*. Then, *f*(*r*) = 0, which can be rearranged to express *r**k* as a linear combination of powers of *r* less than *k*. This equation can be used to reduce away any powers of *r* ≥ *k*. For example, if *f*(*x*) = *x*2 + 1 and *r* is the imaginary unit *i*, then *i*2 + 1=0, or *i*2 = −1. This allows us to define the complex product:

- (
*a*+*bi*)(*c*+*di*) =*ac*+ (*ad*+*bc*)*i*+ (*bd*)*i*2 = (*ac*−*bd*) + (*ad*+*bc*)*i*.

In general, this leads directly to the algebraic number field **Q**, which can be defined as the set of real numbers given by:

*a*_{k−1}*r**k*−1 + ... +*a*_{1}*r*1 +*a*_{0}*r*0, where*a*_{0},...,*a*_{l−1}in**Q**.

The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of *r* ≥ *k* as described above, yielding a value in the same form. To ensure that this field is actually *k*-dimensional and does not collapse to an even smaller field, it is sufficient that *f* is an irreducible polynomial. Similarly, one may define the number field ring **Z** as the subset of **Q** where *a*_{0},...,*a*_{k−1} are restricted to be integers.

Read more about this topic: General Number Field Sieve

### Other articles related to "number fields, fields, number, field, numbers":

...

**Number fields**share a great deal of similarity with another class of

**fields**much used in algebraic geometry known as function

**fields**of algebraic curves over finite

**fields**... in many respects, for example in that

**number**rings are one-dimensional regular rings, as are the coordinate rings (the quotient

**fields**of which is the function

**field**... Therefore, both types of

**field**are called global

**fields**...

... his results is the parametrization of quartic and quintic orders in

**number fields**, thus allowing the study of asymptotic behavior of arithmetic properties of these orders and

**fields**... p-adic analysis, to the study of ideal class groups of algebraic

**number fields**, and to the arithmetic theory of elliptic curves ... density of discriminants of quartic and quintic

**number fields**...

... use the generalized Riemann hypothesis for Dirichlet L-series or zeta functions of

**number fields**rather than just the Riemann hypothesis ... implies that Gauss's list of imaginary quadratic

**fields**with class

**number**1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without using the generalized Riemann hypothesis ... hypothesis implies a weak form of the Goldbach conjecture for odd

**numbers**that every sufficiently large odd

**number**is the sum of three primes, though in 1937 Vinogradov gave an unconditional proof ...

### Famous quotes containing the words fields and/or number:

“Earth has not anything to show more fair:

Dull would he be of soul who could pass by

A sight so touching in its majesty:

This city now doth, like a garment, wear

The beauty of the morning; silent, bare,

Ships, towers, domes, theatres and temples lie

Open unto the *fields* and to the sky;

All bright and glittering in the smokeless air.”

—William Wordsworth (1770–1850)

“As equality increases, so does the *number* of people struggling for predominance.”

—Mason Cooley (b. 1927)