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Magma provides facilities for computing with Galois rings.
The features are currently very basic, but advanced features
will be available in the near future, including support for
the creation of subrings and appropriate embeddings, allowing
lattices of compatible embeddings, just as for finite fields.
A Galois ring R in Magma is considered as a finite algebraic
extension of Z_(p^a) (where p is prime) by a monic
polynomial D∈Z[x] which is irreducible
modulo p. Thus R is presented as the polynomial quotient
ring Z_(p^a)[x]/<D> and is usually written as GR(pa, d), where d is
the degree of D. The cardinality of R is easily seen to be pad.
R has a unique maximal ideal generated by p, and the quotient
ring R/< p > is a finite field isomorphic to
Zp[x]/< D >, where
D is here considered as a polynomial in Zp[x] (the coefficients
are reduced modulo p). This finite field is called the
residue field of R. In the following, we will also
call the integer residue ring Z_(p^a) the base ring of R,
because this is the subring of R generated by 1 and we can
think of R as an extension of Z_(p^a).
For a non-zero element x of R, the valuation of x is defined to be
the largest power of p which divides the coefficients of x,
where x is considered as a polynomial in Zp[x]/< D >. x
is a unit if and only if the valuation of x is zero.
Because of the valuation defined on them, Galois rings are
Euclidean rings, so they may be used in Magma in any place where
general Euclidean rings are valid. This includes many matrix and
module functions, and the computation of Gröbner bases. Linear
codes over Galois rings will be supported in the near future.
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