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

Cardinality

The algebraic numbers include all rational numbers, and both sets of numbers, rational and algebraic, are countable.

Algebraic Properties

The algebraic numbers form a field; in fact, they are the smallest algebraically closed field with characteristic 0.

Degree

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

Examples

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

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

Algebraic numbers via subfields

The field of complex numbers 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 \ \mathbb{C}}   is a linear space over the field of rational numbers 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 \ \mathbb{Q}.}   In this section, by a linear space we will mean a linear subspace of 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 \ \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

$a_{0}\cdot z^{n}+\dots +a_{n-1}\cdot z+a_{n}\ =\ 0$

where $\ a_{0}\neq 0\neq a_{n},$  by $\ a_{n}\cdot z,$  and you quickly get an equality of the form:

$z^{-1}\ =\ b_{0}\cdot z^{n-1}+\cdots +b_{n-1}$

A momentary reflection gives now

Theorem The degree of the inverse $\ z^{-1}$  of any algebraic number $\ z\neq 0$  is equal to the degree of the number $\ z$  itself.

The sum and product of two algebraic numbers

Let $\ 1\in A\subseteq {\mathcal {A}}$  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.