AlgebraicNumber and NumberField #21
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Introduce the concept of AlgebraicNumber and NumberField.
Algebraic numbers are defined to be real or complex numbers, which are zeros of monic polynomials over the rational numbers.
Here each algebraic number
Ais defined by its minimal polynomial and an approximation of the zero of it. The minimal polynomial isthe uniquely determined irreducible monic polynomial, which has this zero. Algebraic operations with algebraic numbers are possible but expensive, because polynomials of a degree of the product of the degrees of the operands have to be factored in the worst case.
A number field over an algebraic number
NF(A)is the set of rational linear combinations of the powers of A. It has dimension of the degree of the minimal polynomial of A. It has a natural field homomorphism with the quotient ring of the minimal polynomial, which is used to allow efficient operations.