Algebraic number

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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 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 monic polynomial with integer coefficients, 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. [1]

Degree and Defining Polynomial

Let ${\displaystyle \ a\in \mathbb {C} }$  be an algebraic number different from ${\displaystyle \ 0.}$  The degree of ${\displaystyle \ a}$  is, by definition, the lowest degree of a polynomial ${\displaystyle \ f,}$  with rational coefficients, for which ${\displaystyle \ f(a)=0.}$ 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 ${\displaystyle \ 1.}$  The rational number a has defining polynomial ${\displaystyle x-a}$. All non-rational algebraic numbers have degree greater than ${\displaystyle \ 1.}$ Note that there are real irrational numbers that are not algebraic (i.e. that are transcendental), such as pi and e.
• ${\displaystyle {\sqrt {2}}}$ 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 ${\displaystyle x^{2}-2}$, it has degree 2, and ${\displaystyle x^{2}-2}$ is its defining polynomial.
• The imaginary unit ${\displaystyle i}$ is an algebraic integer of degree 2, having defining polynomial polynomial ${\displaystyle x^{2}+1}$.
• The golden ratio, ${\displaystyle (1+{\sqrt {5}})/2}$, is also an algebraic number(actually, an integer!) of degree 2, with defining polynomial ${\displaystyle x^{2}-x-1}$.
• If ${\displaystyle a}$ is a rational number, then ${\displaystyle {\sqrt[{n}]{a}}}$ is an algebraic number of degree n, having defining polynomial ${\displaystyle x^{n}-a}$. It is an algebraic integer precisely when a is an integer.

Algebraic numbers via subfields

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

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

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

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

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

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

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

A momentary reflection gives now

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

The sum and product of two algebraic numbers

Let ${\displaystyle \ 1\in A\subseteq {\mathcal {A}}}$  and ${\displaystyle \ 1\in B\subseteq {\mathcal {B}},}$  where ${\displaystyle \ A,B,}$  are finite linear bases of fields ${\displaystyle \ {\mathcal {A}},{\mathcal {B}},}$  respectively. Let ${\displaystyle \ {\mathcal {D}}}$  be the smallest algebra generated by ${\displaystyle \ {\mathcal {A}}\cup {\mathcal {B}}.}$  Then ${\displaystyle \ {\mathcal {D}}}$  is linearly generated by

${\displaystyle \{a\cdot b:\ a\in A\ \land \ b\in B\}}$

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

${\displaystyle \dim({\mathcal {D}})\ \leq \ \dim({\mathcal {A}})\cdot \dim({\mathcal {B}})}$

Now, let ${\displaystyle \ a,b,}$  be arbitrary algebraic numbers of degrees ${\displaystyle \ m,n,}$  respectively. They belong to their respective m- and n-dimensional algebras. The sum and product ${\displaystyle \ a+b,a\cdot b,}$  belong to the algebra generated by the union of the two mentioned algebras. The dimension of the generated algebra is not greater than ${\displaystyle \ m\cdot n.}$ It contains ${\displaystyle \ a+b,a\cdot b,}$  as well as all linear combinations ${\displaystyle \ \alpha \cdot a+\beta \cdot b,}$  with rational coefficients ${\displaystyle \ \alpha ,\beta .}$  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 mn. 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.

Notes

1. If 1 + 1 = 0 in the field, the characteristic is said to be 2; if 1 + 1 + 1 = 0 the characteristic is said to be 3, and forth. If there is no ${\displaystyle n}$ such that adding 1 ${\displaystyle n}$ times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite.