A number field order in a number field of extension degree over is a subring of the ring of integers of containing 1 and forming a free Abelian group of rank over .

The ring of integers of every number field is an order, known as the maximal order, of . Every order of is contained in the maximal order. If is an algebraic integer with , then is an order of , though it need not be maximal.

The multiplier order of a full lattice is . Its absolute discriminant is , where is the number field discriminant and is the ring of integers.

See also

Algebraic Integer, Extension Field Degree, Field Order, Number Field, Number Field Discriminant, Order, Ring, Ring of Integers

Portions of this entry contributed by David Terr

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References

Einsiedler, M.; Lindenstrauss, E.; Michel, P.; and Venkatesh, A. "Distribution of Periodic Torus Orbits and Duke's Theorem for Cubic Fields." Ann. Math. 173, 815-885, 2011. https://doi.org/10.4007/annals.2011.173.2.5.Milne, J. S. "Algebraic Number Theory." Version 3.08, 19 Jul 2020. https://www.jmilne.org/math/CourseNotes/ANT.pdf.

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Number Field Order

Cite this as:

Weisstein, Eric W., with contributions by David Terr. "Number Field Order." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NumberFieldOrder.html

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