Algebraic Numbers

Let \(F\) and \(K\) be fields with \(F \subseteq K\). A number \(\alpha \in K\) is said to be algebraic over \(F\) if there exists a polynomial \(p(x) \in F[x]\) such that \(f(\alpha) = 0\). If \(\alpha \in K\) is algebraic over \(F\) then the subset \(I\) of \(F[x]\) given by the set of all polynomials \(p(x) \in F[x]\) satisfying \(f(\alpha) = 0\) is an ideal of \(F[x]\). \[I := \{p(x) \in F[x] : p(\alpha) = 0\} \text{ is an ideal of } F[x]\] \(F[x]\) is a principal ideal domain and the unique monic polynomial \(m(x) \in F[x]\) which generates \(I\) is called the minimal polynomial of \(\alpha\).

Algebraon supports the following algebraic numbers:

• Real numbers \(\alpha \in \mathbb{R}\) which are algebraic over \(\mathbb{Q}\). Represented by the minimal polynomial \(m(x) \in \mathbb{Q}[x]\) of \(\alpha\) and a rational isolating interval \((a, b) \subseteq \mathbb{R}\) consisting a pair of rational numbers \(a, b \in \mathbb{Q}\) such that \(\alpha \in (a, b)\) and no other root of \(m(x)\) is contained in \((a, b)\).
• Complex numbers \(\alpha \in \mathbb{C}\) which are algebraic over \(\mathbb{Q}\). Represented by the minimal polynomial \(m(x) \in \mathbb{Q}[x]\) of \(\alpha\) and a rational isolating box in the complex plane.
• For a prime \(p \in \mathbb{N}\), \(p\)-adic numbers \(\alpha \in \mathbb{Q}_p\) which are algebraic over \(\mathbb{Q}\). Represented by the minimal polynomial \(m(x) \in \mathbb{Q}[x]\) of \(\alpha\) and a \(p\)-adic isolating ball consisting of a rational number \(c \in \mathbb{Q}\) and a valuation \(v \in \mathbb{N} \sqcup {\infty}\) such that \(\alpha - c\) has \(p\)-adic valuation at least \(v\) and no other root \(\beta\) of \(m(x)\) is such that \(\beta - c\) has valuation \(p\)-adic at least \(v\).
• If \(m(x) \in \mathbb{Q}[x]\) is irreducible then the quotient field \(K := \frac{\mathbb{Q}[x]}{m(x)}\) is a finite dimensional extension of \(\mathbb{Q}\), also known as an algebraic number field. Every element of \(K\) is algebraic over \(\mathbb{Q}\). Elements of \(K\) are represented by rational polynomials up to adding rational polynomial multiples of \(m(x)\).