An algebraically closed field is based on an affine algebra (or quotient ring of a multivariate polynomial ring by an ideal of "relation" polynomials). The defining polynomials of this affine algebra are not necessarily irreducible -- the system avoids factorization over an algebraic number field when possible, and automatically splits the defining polynomials of the affine algebra when factors are found during computations with the field. These factors often arise automatically because of the structure of the algorithm which is computing over the field.
Many technical optimizations have been designed to make the system practical. For example, the most expensive arithmetic operation by far in the whole system is, quite surprisingly, the testing of whether an element of the field is zero or not (which is utterly trivial for most other rings, of course)! To allow for the parallelism amongst conjugates to work, Magma performs a recursive GCD computation with the element, considered as a polynomial in its highest variable, and the appropriate defining polynomial, to determine the result. If the GCD is non-trivial, then this forces a splitting of the defining polynomial, all elements of the field are reduced, and the original element may now be zero. More details concerning the internal design of the system can be found in [Ste10].
Care must be taken with the interpretation of the roots of a polynomial in this system. The roots of polynomials are only defined algebraically, and the user may wish to identify them with some particular elements of the complex field, for example, but one cannot assume that the system will follow the embedding one wishes. An example will demonstrate. Suppose that α is a root of x2 - 2, β is a root of x2 - 3, and γ is a root of x2 - 6. Does γ = α.β or does γ = - α.β? The system will have to make a choice between the two possibilities if the situation arises, but the choice which it will make cannot be predicted beforehand. That is, even if we might like to interpret things as α=√2, β=√3 and γ=√6 (referring to the positive real roots in each case), it cannot be assumed that this will hold in the particular algebraically closed field, since the roots are only defined algebraically.
In the following descriptions, the adjective invariable, applied to a value, means that the mathematical value of a function will not change, despite simplifications, etc. which may occur after the function is called. That is, the value returned by the function may be considered constant with respect to the eq operator. A value may print differently later (because of simplifications of the field), but the mathematical value will never change.
In contrast, the adjective variable means that the mathematical value of a function may change (as a result of simplifications). This is usually only a property of access-like functions like Degree(A, v) (see below). But the fact that most functions below have invariable return values enables the illusion of a true field to be sustained.
Separate algebraic fields should be created for separate problems which involve root taking (if the roots of the different problems are unrelated). Otherwise it may be unnecessarily expensive to compute in the field with roots of unrelated polynomials.
Currently, the base field of an algebraic closure may be a finite field, the rational field Q, or a rational function field over a finite field or Q.