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A ring element can often be coerced into a ring other than its parent.
The need for this occurs for example when one wants to perform a binary ring
operation on
elements of different structures, or when an intrinsic function is
invoked on elements for which it has not been defined.
The basic principle is that such an operation may be performed whenever it
makes sense mathematically. Before the operation can be performed however,
an element may need to be coerced into some structure in which
the operation can legally be performed.
There are two types of coercion: automatic and forced
coercion. Automatic coercion occurs when Magma can figure out for
itself what the target structure should be, and how elements of the
originating structure can be coerced into that structure.
In other cases Magma may still be able to perform the coercion, provided
the target structure has been specified; for this type of coercion
R ! x instructs Magma to execute the coercion of element x
into ring R.
The precise rules for automatic and forced coercion between rings are
explained in the next two subsections.
It is good to keep an important general distinction between automatic
and forced coercion in mind: whether or not automatic coercion will succeed
depends on the originating and the target structure alone,
while for forced coercion success may
depend on the particular element as well. Thus, integers can be
lifted automatically to the rationals if necessary, but conversely,
only the integer elements of Q can be coerced into Z by using !.
The subsections below will describe for specific rings R and S in Magma
whether or not an element r of R can be lifted automatically or by
force into S. Suppose that the unary Magma function Function takes
elements of the type of S as argument and one is interested in the result
of that function when applied to r. If R can be coerced automatically
into a unique structure S of the desired type,
then Function(r) will produce the required result. If
R cannot be coerced automatically into such S, but forced coercion on r is
possible, then Function(S ! r) will yield the desired effect.
If, finally, neither automatic nor forced coercion is possible, it may be possible to
define a map m from R to S, and then Function( m(r) ) will
give the answer.
For example, the function Order is defined for
elements of residue class rings (among others). But Order(3) has no obvious
interpretation (and an error will arise) because there is not a unique
residue class ring in which this should be evaluated.
If a binary operation  : C x C -> C
on members C of a category of rings (C) is applied to elements
r and s of members R and S from (C), the same rules for
coercion will be used to determine the legality of r s. If
s can be coerced automatically into R, then this will take
place and r s will be evaluated in R; otherwise, if
r can be coerced automatically into S, then
r s will be evaluated in S.
If neither succeeds, then, in certain cases, Magma will try to find
an existing common overstructure T for R and S, that is,
an object T from (C) such that ⊂T⊃S; then both
r and s will be coerced into T and the result t=r s will be
returned as an element of T. If none of these cases apply,
an error results. It may still be possible to evaluate
by forced coercion of r into S or s into R, using
(S ! r) o s or using r o (R ! s).
We will first deal with the easier of the two cases: automatic coercion.
A simple demonstration of the desirability of automatic coercion
is given by the following example:
1 + (1/2);
It is obvious that one wants the result to be 3/2: we want to
identify the integer 1 with the rational number 1 and perform the
addition in Q, that is, we clearly wish to have automatic coercion
from Z to Q.
The basic rule for automatic coercion is:
automatic coercion will only take place when there
exists a unique target structure and an obvious
homomorphism from the parent structure to the target structure
In particular, if one structure is naturally
contained in the other (and Magma knows about it!),
automatic coercion will take place. (The provision
that Magma must know about the embedding is in particular relevant for
finite fields and number fields; in these cases it is possible to create
subrings, or even isomorphic rings/fields, for which the embedding is not
known.)
Also, for any ring R there is a natural ring homomorphism Z -> R,
hence any integer can be coerced automatically into any ring.
See the (pdf) Handbook for a table detailing the rules for automatic coercion.
It is appropriate here to make some general remarks only.
Integers will be automatically coerced into any ring, and rationals
into any ring except a finite field or a residue class ring.
Operations involving elements of real or complex fields of different
precisions automatically coerce the numbers into the field with the
higher precision.
Elements of different finite field elements can be
operated upon with automatic coercion only if one field is a subset
of the other or both fields have been created inside another finite field.
In addition to these rules, general rules apply to
polynomial and matrix algebras. The rules for polynomial rings are
as follows.
An element s from a ring S can be automatically coerced
into R[X1, ..., Xn] if
either S equals R[X1, ..., Xi] for some 1≤i≤n,
or S=R. Note that in the latter case the element s must be an element of the
coefficient ring R, and that it is not sufficient for it
to be coercible into R.
So, for example, we can add an integer and a polynomial over the
integers, we can add an element f of Z[X] and g of
Z[X, Y], but not an integer and a polynomial
over the rationals.
An element s can be coerced automatically
in the matrix ring Mn, n(R) if it is coercible automatically into the coefficient ring R, in which case s will
be identified with the diagonal matrix that has each diagonal entry equal to s.
So we can add an integer and a matrix ring element
over the rationals, but we cannot automatically add
elements of Mn, n(Z) and
Mn, n(Q), nor elements from M2, 2(Z) and M3, 3(Z).
In certain cases where automatic coercion will not take place, one can
cast an element into the ring in which the operation should take place.
If, for example, one is working in a ring Z/pZ, and p happens
to be prime, it may occur that one wishes to perform some finite
field operations on an element in the ring; if F is a finite field
of characteristic p an element x of Z/pZ can be cast into
an element of F using
F ! x;
See the (pdf) Handbook for a table detailing the rules for non-automatic coercion.
It is appropriate here to make some general remarks only.
Non-automatic coercion may only apply
to certain elements of the domain; thus only those elements of Q can be
coerced into Z that are in fact integers. Similarly, if an element
of a quadratic field with a negative discriminant is to be
coerced into a real field then it must correspond to a
complex number with no imaginary part.
An element from Fps can only be coerced
into Z/mZ if s=1 and m=p.
Any element from Fps
can be coerced into Fpr if s divides r, and otherwise
only a subset of the field is coercible. Similarly for cyclotomic
fields.
The rules for coercion from and to polynomial rings and matrix rings
are as follows.
If an attempt is made to forcibly coerce s into P=R[X1, ..., Xn],
the following steps are executed successively:
- (a)
- if s is an element of P it remains unchanged;
- (b)
- if s is a sequence, then the zero element of P is returned
if s is empty, and if it is non-empty but the elements of the sequence
can be coerced into P[X1, ..., Xn - 1] then the polynomial equaling the sum over j of s[j]Xnj - 1
is returned;
- (c)
- if s can be coerced into the coefficient ring R, then
the constant polynomial s is returned;
- (d)
- if s is a polynomial in R[X1, ..., Xk] for some
1≤k≤n, then it is lifted in the obvious way into P;
- (e)
- if s is a polynomial in R[X1, ..., Xk] for some
k > n, but constant in the indeterminates Xn + 1, ..., Xk,
then s is projected down in the obvious way to P.
If none of these steps successfully coerces s into P, an error
occurs.
The ring element s can be coerced into Mn, n(R) if
either it can be coerced into R (in which case s will
be identified with the diagonal matrix with s on the diagonal), or
s∈S=Mn, n(R'), where R' can be coerced into R.
Also a sequence of n2 elements coercible into R can be coerced
into the matrix ring Mn, n(R).
Elements from a matrix ring Mn, n(R) can only be coerced into rings other
than a matrix ring if n=1; in that case
the usual rules for the coefficient ring R apply.
Note that in some cases it is possible to go from (a subset of) some
structure to another in two steps, but not directly: it is possible to
go
> y := L ! (Q ! x)
to coerce a rational element of one number field into another via the rationals.
Finally we note that the binary Boolean operator in returns true if
and only if forced coercion will be successful.
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