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:
- ak−1rk−1 + ... + a1r1 + a0r0, where a0,...,al−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,...,ak−1 are restricted to be integers.
Read more about this topic: General Number Field Sieve
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