Algebraic number

In mathematics, and more specifically—in number theory, an algebraic number is a complex number that is a root of a polynomial with rational coefficients. Real or complex numbers that are not algebraic are called transcendental numbers.

Instances of algebraic numbers have been studied for for millennia as solutions of quadratic equations. They appear indirectly in the cakravāla method from the 11th century. In the 15th century, they arose in finding general solutions to cubic and quartic equations. However, the properties of algebraic numbers were not intensively studied until algebraic numbers appeared in an attempt to solve Fermat's last theorem.

The theory of algebraic numbers that ensued forms the foundation of modern algebraic number theory. Algebraic number theory is now an immense field, and one of current research, but so far has found few applications to the physical world.

Alternative Characterization

Every polynomial with rational coefficients can be converted to one with integer coefficients by multiplying through by the least common multiple of the denominators of the coefficients. It follows that the term "algebraic number" can also be defined as a 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.

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 and Defining Polynomial

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.} There is a unique monic polynomial of degree d having a as a root. It is the defining polynomial (or minimal polynomial) for a.

Examples

• Rational numbers 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.}   The rational number a has defining 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-a } . 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.} Note that there are real irrational numbers that are not algebraic (i.e. that are transcendental), such as pi and e.

• 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. It is not rational, so must have degree greater than 1. 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} , it has degree 2, 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 x^2-2} is its defining polynomial.

• 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, having defining polynomial 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} .

• The golden ratio, 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+\sqrt{5})/2 } , is also an algebraic number(actually, an integer!) of degree 2, with defining 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-x-1 } .

• If 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 a rational number, 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 \sqrt[n]{a} } is an algebraic number of degree n, having defining 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^n-a } . It is an algebraic integer precisely when a is an integer.

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 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};}   and by algebra we mean a linear space which is closed under the multiplication, and which has 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}   as its element. The following properties of a complex 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 \in \mathbb{C}}   are equivalent:

• 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}   is an algebraic number 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 \ \le n;}
• 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}   belongs to an algebra of linear dimension 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 \ \le n.}

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

Actually, every finite dimensional algebra 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\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 $\ a+b,a\cdot b,$  belong to the algebra generated by the union of the two mentioned algebras. The dimension of the generated algebra is not greater than $\ m\cdot n.$ It contains $\ a+b,a\cdot b,$  as well as all linear combinations $\ \alpha \cdot a+\beta \cdot b,$  with rational coefficients $\ \alpha ,\beta .$  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.