EMBEDDING OF AN ORDER INTO A MAXIMAL ORDER
Horst G. Zimmer

yields an order om+1 properly containing om . As soon as an intermediate orderoh in the above tower is found for which there is no prime p with p2 | dh such that
[Rp,h /Rp,h ] properly contains oh , the desired maximal order(Springer LNM 262 [1972] pp 25–27)

Let there be given a finite dimensional commutative algebra A of degree n over
Q and an order o in A such thatA = o ⊗Z Q.
It is our goal to find the maximal order O in A containing o. We assume theorder o in A to be defined by means of a basis ω1 , ω2 , . . . , ωn over Z together
with a multiplication table. All orders in A considered in this section are also
supposed to be given by a Z-basis that is expressed in terms of the basis ω1 ,
ω2 , . . . , ωn of o.
The task proposed was first solved by Berwick (1927). Subsequently, Zassenhaus (1967) developed a more general algorithm which was implemented in theform of an effective computer program. The basic idea is to build up a tower ofintermediate orderso ⊂ o1 ⊂ · · · ⊂ om ⊂ om+1 ⊂ · · · ⊂ O
so as to reach O after a finite number of steps.
A remark concerning effectiveness is in order. The maximal order O over o soughtcannot be effectively constructed by a search procedure which finds O among all(finitely many) orders between o and d−1 o where d is the discriminant of o.
Rather we proceed in the following fashion. Let us assume that we have arrived
already at an intermediate order om and that om 6= O. We wish to constructom+1 . To this end we introduce for a rational prime p the p-radical of o, Rp,m ,that is, the ideal containing pom which is defined by
Rp,m = x ∈ om x mod pom is nilpotent .
The construction of om+1 is now based on the following

O = ohhas been reached.
Note that the discriminants dm and dm+1 of the orders om and om+1 respectively
are subject to the relation
dm = (det Tm )2 dm+1−1
where Tmis the n × n transition matrix over Z by which a Z-basis of om+1 isexpressed in terms of a Z-basis of om . From this remark it can also be seen that
there is only a finite number of candidates om competing for O.

In order to obtain a Z-basis of om+1 from a Z-basis of om we use the p-trace
radical of om , that is, the idealTp,m = x ∈ om tr(xom ) ⊆ pZ ,where tr denotes the trace function belonging to the regular representation of om .
We have thenpom ⊆ Rp,m ⊆ Tp,mwith the equality sign standing on the right hand side whenever p > n. Using
these inclusions we first determine a suitable Z-basis for Tp,m in terms of a Zbasis of om and then derive from it a Z-basis for Rp,m and, finally, a Z-basis for[Rp,m /Rp,m ]. All this is accomplished by matrix operations involving matrices
with entries in Z.
The effectiveness of this algorithm can be further improved by dealing simultaneously rather than separately with those primes p for which
p2 | dandp > n.

Theorem. Suppose that om 6= O. Then there exists a prime p, whose square
divides the discriminant dm of the order om , such that the om -quotient module[Rp,m /Rp,m ] is an order in A that contains om properly.

Specifically, on defining the number

Here, the quotient module [Rp,m /Rp,m ] consists of those elements in A which
multiply Rp,m into itself.

we introduce the ideals Rd0 ,m , Td0 ,m instead of Rp,m , Tp,m respectively andproceed in an analogous manner to that outlined above.

On the grounds of this theorem we know that, amongst all primes p with p2 | dm ,there can be found a prime (the smallest such) for which the definition

The algorithm is naturally of particular interest in the case in which A = K is afinite algebraic number field over Q because it facilitates the construction of thering O of all algebraic integers in K.

om+1 = [Rp,m /Rp,m ]d0 =Q

p2 |d, p>n p22-May-2013, 9:35 pm