Contents - Student subdomain for University of Bath

6.1. THE D5 APPROACH TO ALGEBRAIC NUMBERS 167
and this is purely a polynomial in α.
In the case of Figure 6.1, f(t) = t 2 − 1 and p(t) = 1, q(t) = t − 1. The gcd is
√
1
indeed 1, with f = −1, d = t + 1. Hence √
2−1
= 2+1
( √ 2+1)( √ = √ 2 + 1. Such
2−1)
is the simplicity of this approach that it is usual to insist that f is irreducible,
however in section 6.1 we will explain an alternative approach.
Definition 79 If f in Definition 77 or 78 is irreducible and primitive, we say
that f is the minimal polynomial of α. Strictly speaking, we have only defined
f up to associates, but this is usually ignored in theory, though it can be tedious
in practice: see the description of the canonicalUnitNormal property in [DT90]
for a pragmatic solution.
Open Problem 14 (Algebraic Numbers Reviewed) Reconsider the standard
approach to factoring polynomials with algebraic number/function coefficients,
as described above, in the light of recent progress in factoring polynomials
with integer coefficients, notably [vH02].
6.1 The D5 approach to algebraic numbers
[DDDD85] TO BE COMPLETED