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 rk. 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 = (acbd) + (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 rk 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

Famous quotes containing the words number and/or fields:

    The growing good of the world is partly dependent on unhistoric acts; and that things are not so ill with you and me as they might have been, is half owing to the number who lived faithfully a hidden life, and rest in unvisited tombs.
    George Eliot [Mary Ann (or Marian)

    Nature will not let us fret and fume. She does not like our benevolence or our learning much better than she likes our frauds and wars. When we come out of the caucus, or the bank, or the abolition-convention, or the temperance-meeting, or the transcendental club, into the fields and woods, she says to us, “so hot? my little Sir.”
    Ralph Waldo Emerson (1803–1882)