Algebraic number

In mathematics, and more specifically—in number theory, an algebraic number is any complex number that is a root of a polynomial with rational coefficients. Any polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators, and every complex root of a polynomial with integer coefficients is an algebraic number. If an algebraic number x can be written as the root of a polynomial with integer coefficients which is also monic, that is, one whose leading coefficient is 1, then x is called an algebraic integer.

The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are countable. The algebraic numbers form a field; in fact, they are the smallest algebraically closed field with characteristic 0.

Real or complex numbers that are not algebraic are called transcendental numbers.

Degree

Let $\ a\in \mathbb {C}$  be an algebraic number different from $\ 0.$  The degree of $\ a$  is, by definition, the lowest degree of a polynomial $\ f,$  with rational coefficients, for which $\ f(a)=0.$

Examples

Rational numbers different from $\ 0$  are algebraic and of degree $\ 1.$  All non-rational algebraic numbers have degree greater than $\ 1.$

${\sqrt {2}}$ is an algebraic number of degree 2, and, in fact, an algebraic integer, as it is a root of the polynomial $x^{2}-2$. Similarly, the imaginary unit $i$ is an algebraic integer of degree 2, being a root of the polynomial $x^{2}+1$.

Algebraic numbers via subfields

The field of complex numbers $\ \mathbb {C}$  is a linear space over the field of rational numbers $\ \mathbb {Q} .$  In this section, by a linear space we will mean a linear subspace of $\ \mathbb {C}$  over $\ \mathbb {Q} ;$  and by algebra we mean a linear space which is closed under the multiplication, and which has $\ 1$  as its element. The following properties of a complex number $\ z\in \mathbb {C}$  are equivalent:

• $\ z$  is an algebraic number of degree $\ \leq n;$
• $\ z$  belongs to an algebra of linear dimension $\ \leq n.$

Indeed, when the first condition holds, then the powers $\ 1,z,\dots ,z^{n-1}$  linearly generate the algebra required by the second condition. And if the second condition holds then the $\ (n+1)$  elements $1,z,\dots ,z^{n}$  are linearly dependent (over rationals).

Actually, every finite dimensional algebra $\ A\subseteq \mathbb {C}$  is a field—indeed, divide an equality

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0\cdot z^n + \dots+ a_{n-1}\cdot z + a_n\ =\ 0}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a_0\ne 0\ne a_n,}   by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a_n\cdot z,}   and you quickly get an equality of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{-1}\ =\ b_0\cdot z^{n-1}+\cdots + b_{n-1}}

A momentary reflection gives now

Theorem The degree of the inverse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ z^{-1}}   of any algebraic number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ z\ne 0}   is equal to the degree of the number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ z}   itself.

The sum and product of two algebraic numbers

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1 \in A\subseteq \mathcal A}   and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1 \in B\subseteq \mathcal B,}   where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ A,B,}   are finite linear bases of fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal A,\mathcal B,}   respectively. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal D}   be the smallest algebra generated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal A\cup \mathcal B.}   Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal D}   is linearly generated by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{a\cdot b :\ a\in A\ \and\ b\in B\}}

Thus the linear dimensions (over rationals) of the three algebras satisfy inequality:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim(\mathcal D)\ \le\ \dim(\mathcal A)\cdot \dim(\mathcal B)}

Now, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,b,}   be arbitrary algebraic numbers of degrees Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ m,n,}   respectively. They belong to their respective m- and n-dimensional algebras. The sum and product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a+b, a\cdot b,}   belong to the algebra generated by the union of the two mentioned algebras. The dimension of the generated algebra is not greater than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ m\cdot n.} It contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a+b, a\cdot b,}   as well as all linear combinations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \alpha\cdot a + \beta\cdot b,}   with rational coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \alpha,\beta.}   This proves:

Theorem  The sum and the product of two algebraic numbers of degree m and n, respectively, are algebraic numbers of degree not greater than m•n. The same holds for the linear combinations with rational coefficients of two algebraic numbers.

As a corollary to the above theorem, together with the previous section, we obtain:

Theorem  The algebraic numbers form a field.