Algebraic numbers

Algebraic numbers

An algebraic number is a real number that can be expressed as the root of a non-zero polynomial in one variable with rational coefficients. Specifically, an algebraic number of degree n is a root of a monic polynomial in x of degree n and is not the root of any polynomial in x of degree less than n. The minimal polynomial of an algebraic number is the polynomial of the least degree with rational coefficients that has the algebraic number as a root, and the degree of the minimal polynomial is called the degree of the algebraic number.From: A game theoretical approach for finding near-optimal solutions of an optimization problem [2022], Algebraic Number Theory Second [2011], Geometry of symplectic partially hyperbolic automorphisms on 4-torus [2020]

Integral Domains, Ideals, and Unique Factorization

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Published in Richard A. Mollin, Algebraic Number Theory Second, 2011

Richard Mollin

An algebraic number, α, of degree d∈N is a root of a monic polynomial in Q[x] of degree d and not the root of any polynomial in Q[x] of degree less than d. In other words, an algebraic number is the root of an irreducible polynomial of degree d over Q. Denote the subfield of C consisting of all algebraic numbers by N‾, and the set of all algebraic integers in N‾ by A. An algebraic number field, or simply number field, is of the form F=Qα1,α2,…,αn⊆Cwithn∈Nwhereαj∈N‾forj=1,2,…,n

Geometry of symplectic partially hyperbolic automorphisms on 4-torus

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Published in Dynamical Systems, 2020

L. M. Lerman, K. N. Trifonov

Recall a number be algebraic if it is a root of a polynomial with rational coefficients, the number ξ is called integer algebraic, if it is a root of a polynomial with integer coefficients whose leading coefficient equals to unity (a monic polynomial). With any algebraic number, the notion of its degree is associated. Namely, if the number ξ is the root of a polynomial of some degree, then multiplying this polynomial at another polynomial with rational coefficients we obtain a polynomial of a greater degree with the root ξ. The polynomial of the least degree with rational coefficients having the root ξ is called the minimal polynomial of the algebraic number ξ, and its degree is called the degree of the number ξ. In particular, all rational numbers are algebraic numbers of degree one. The polynomial Q above has degree 4 due to the following theorem ([17], theorem 3.3.14, p . 147)

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Integral Domains, Ideals, and Unique Factorization

Geometry of symplectic partially hyperbolic automorphisms on 4-torus