General Number Field Sieve - Number Fields

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 rk 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) = x2 + 1 and r is the imaginary unit i, then i2 + 1=0, or i2 = −1. This allows us to define the complex product:

(a+bi)(c+di) = ac + (ad+bc)i + (bd)i2 = (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−1rk−1 + ... + a₁r1 + a₀r0, where a₀,...,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 a₀,...,a_k−1 are restricted to be integers.