Editing General number field sieve (section)

== Number fields ==
{{main|Number field}}
Suppose {{mvar|f}} is a {{mvar|k}}-degree polynomial over <math display=inline>\mathbb Q</math> (the rational numbers), and {{mvar|r}} is a complex root of {{mvar|f}}. Then, {{math|''f''(''r'')&nbsp;{{=}}&nbsp;0}}, which can be rearranged to express {{math|''r''<sup>''k''</sup>}} as a linear combination of powers of {{mvar|r}} less than {{mvar|k}}. This equation can be used to reduce away any powers of {{math|''r''}} with exponent {{math| ''e'' ≥ ''k''}}. For example, if {{math|''f''(''x'')&nbsp;{{=}}&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;1}} and {{mvar|r}} is the imaginary unit {{mvar|i}}, then {{math|''i''<sup>2</sup>&nbsp;+&nbsp;1&nbsp;{{=}}&nbsp;0}}, or {{math|''i''<sup>2</sup>&nbsp;{{=}}&nbsp;−1}}. This allows us to define the complex product:
:<math>
\begin{align}
(a+bi)(c+di) & = ac + (ad+bc)i + (bd)i^2 \\[4pt]
& = (ac - bd) + (ad+bc)i.
\end{align}
</math>
In general, this leads directly to the [[algebraic number field]] <math display=inline>\mathbb Q[r]</math>, which can be defined as the set of [[complex number]]s given by:

:<math>a_{k-1}r^{k-1} + \cdots + a_1 r^1 + a_0 r^0, \text{ where } a_0,\ldots,a_{k-1} \in \mathbb Q.</math>

The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of {{math|''r''}} with exponent {{math| ''e'' ≥ ''k''}} as described above, yielding a value in the same form. To ensure that this field is actually {{mvar|k}}-dimensional and does not collapse to an even smaller field, it is sufficient that {{mvar|f}} is an [[irreducible polynomial]] over the rationals. Similarly, one may define the [[ring of integers]] <math display=inline> \mathbb O_{\mathbb Q[r]} </math> as the subset of <math display=inline>\mathbb Q[r]</math> which are roots of [[monic polynomial|monic polynomials]] with integer coefficients. In some cases, this ring of integers is equivalent to the ring <math display=inline> \mathbb Z[r] </math>. However, there are many exceptions.<ref name="AlgNumbersRibenboim">{{cite book | title=Algebraic Numbers | publisher=Wiley-Interscience | author=Ribenboim, Paulo | year=1972 | isbn=978-0-471-71804-8}}</ref>