**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

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