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   be an algebraic number different from   The degree of   is, by definition, the lowest degree of a polynomial   with rational coefficients, for which 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   The rational number a has defining polynomial . All non-rational algebraic numbers have degree greater than Note that there are real irrational numbers that are not algebraic (i.e. that are transcendental), such as pi and e.
  • 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 , it has degree 2, and is its defining polynomial.
  • The imaginary unit is an algebraic integer of degree 2, having defining polynomial polynomial .
  • The golden ratio, , is also an algebraic number(actually, an integer!) of degree 2, with defining polynomial .
  • If is a rational number, then is an algebraic number of degree n, having defining polynomial . It is an algebraic integer precisely when a is an integer.

Algebraic numbers via subfields

The field of complex numbers   is a linear space over the field of rational numbers   In this section, by a linear space we will mean a linear subspace of   over   and by algebra we mean a linear space which is closed under the multiplication, and which has   as its element. The following properties of a complex number   are equivalent:

  •   is an algebraic number of degree
  •   belongs to an algebra of linear dimension

Indeed, when the first condition holds, then the powers   linearly generate the algebra required by the second condition. And if the second condition holds then the   elements   are linearly dependent (over rationals).

Actually, every finite dimensional algebra   is a field—indeed, divide an equality

where   by   and you quickly get an equality of the form:

A momentary reflection gives now

Theorem The degree of the inverse   of any algebraic number   is equal to the degree of the number   itself.

The sum and product of two algebraic numbers

Let   and   where   are finite linear bases of fields   respectively. Let   be the smallest algebra generated by   Then   is linearly generated by

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

Now, let   be arbitrary algebraic numbers of degrees   respectively. They belong to their respective m- and n-dimensional algebras. The sum and product   belong to the algebra generated by the union of the two mentioned algebras. The dimension of the generated algebra is not greater than It contains   as well as all linear combinations   with rational coefficients   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 such that adding 1 times gives 0, we say the characteristic is 0. A field of positive characteristic need not be finite.