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 ![Z[alpha]](/images/equations/NumberFieldOrder/Inline13.svg)
is an order of 
, though it need not be maximal.
The multiplier order of a full lattice 
is 
. Its absolute discriminant is ![|Disc(K)|[O_K:O(M)]^2](/images/equations/NumberFieldOrder/Inline17.svg)
, where 
is the number
field discriminant and 
is the ring of integers.
