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.
- 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
Other articles related to "number fields, fields, number, numbers, field":
... applications use the generalized Riemann hypothesis for Dirichlet L-series or zeta functions of number fields rather than just the Riemann hypothesis ... the generalized Riemann hypothesis implies that Gauss's list of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner ... 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 ...
... 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 ... 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 ... Therefore, both types of field are called global fields ...
... parametrization of quartic and quintic orders in number fields, thus allowing the study of asymptotic behavior of arithmetic properties of these orders and fields ... problems and p-adic analysis, to the study of ideal class groups of algebraic number fields, and to the arithmetic theory of elliptic curves ... of discriminants of quartic and quintic number fields ...
Famous quotes containing the words fields and/or number:
“Gone are the days when my heart was young and gay,
Gone are my friends from the cotton fields away,
Gone from the earth to a better land I know,
I hear their gentle voices calling Old Black Joe.”
—Stephen Collins Foster (18261864)
“Even in ordinary speech we call a person unreasonable whose outlook is narrow, who is conscious of one thing only at a time, and who is consequently the prey of his own caprice, whilst we describe a person as reasonable whose outlook is comprehensive, who is capable of looking at more than one side of a question and of grasping a number of details as parts of a whole.”
—G. Dawes Hicks (18621941)