Editing Discriminant (section)

===Discriminant of an algebraic number field===
{{main article|Discriminant of an algebraic number field}}The discriminant of an [[algebraic number field]] measures the size of the ([[ring of integers]] of the) algebraic number field. 

More specifically, it is proportional to the squared volume of the [[fundamental domain]] of the [[ring of integers]], and it regulates which [[Prime number|primes]] are [[Ramified prime#In algebraic number theory|ramified]].

The discriminant is one of the most basic invariants of a number field, and occurs in several important [[Analytic Number Theory|analytic]] formulas such as the [[Functional equation (L-function)|functional equation]] of the [[Dedekind zeta function]] of ''K'', and the [[analytic class number formula]] for ''K''. [[Hermite–Minkowski theorem|A theorem]] of [[Charles Hermite|Hermite]] states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an [[open problem]], and the subject of current research.<ref>{{Citation| last1=Cohen| first1=Henri| author-link=Henri Cohen (number theorist)| last2=Diaz y Diaz| first2=Francisco | last3=Olivier| first3=Michel| contribution=A Survey of Discriminant Counting| title=Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002| editor-last=Fieker| editor-first=Claus| editor2-last=Kohel| editor2-first=David R.| publisher=Springer-Verlag| location=Berlin| series=Lecture Notes in Computer Science | issn=0302-9743| isbn=978-3-540-43863-2| doi=10.1007/3-540-45455-1_7| year=2002| volume=2369| pages=80–94| mr=2041075}}</ref>

Let ''K'' be an algebraic number field, and let ''O<sub>K</sub>'' be its [[ring of integers]]. Let ''b''<sub>1</sub>, ..., ''b<sub>n</sub>'' be an [[integral basis]] of ''O<sub>K</sub>'' (i.e. a basis as a [[Module (mathematics)|'''Z'''-module]]), and let {σ<sub>1</sub>, ..., σ<sub>''n''</sub>} be the set of embeddings of ''K'' into the [[Complex number|complex numbers]] (i.e. [[injective]] [[Ring homomorphism|ring homomorphisms]] ''K''&nbsp;→&nbsp;'''C'''). The '''discriminant''' of ''K'' is the [[Square (algebra)|square]] of the [[determinant]] of the ''n'' by ''n'' [[Matrix (mathematics)|matrix]] ''B'' whose (''i'',''j'')-entry is σ<sub>''i''</sub>(''b<sub>j</sub>''). Symbolically,

: <math>\Delta_K=\det\left(\begin{array}{cccc}
\sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\
\sigma_2(b_1) & \ddots & & \vdots \\
\vdots & & \ddots & \vdots \\
\sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n)
\end{array}\right)^2.
</math>


The discriminant of ''K'' can be referred to as the absolute discriminant of ''K'' to distinguish it from the  of an [[Field extension|extension]] ''K''/''L'' of number fields. The latter is an [[Ideal (ring theory)|ideal]] in the ring of integers of ''L'', and like the absolute discriminant it indicates which primes are ramified in ''K''/''L''. It is a generalization of the absolute discriminant allowing for ''L'' to be bigger than '''Q'''; in fact, when ''L''&nbsp;=&nbsp;'''Q''', the relative discriminant of ''K''/'''Q''' is the [[principal ideal]] of '''Z''' generated by the absolute discriminant of ''K''.