ALGEBRAIC NUMBERS

Abstract. The article discusses the properties, signs and distinctive features of algebraic numbers.

Keywords: algebraic numbers; greatest common divisor; least common multiple; problem solving.

The purpose of this article is to understand the essence of algebraic numbers. Its tasks: to define the concept of algebraic numbers, to consider properties, signs and distinctive features, and also to solve problems on the topic of algebraic numbers.

There is the following definition of an algebraic number: "The roots of non-zero polynomials with rational coefficients are called algebraic numbers."

And also integer algebraic numbers are called the roots of non-zero polynomials with integer coefficients and the highest coefficient.

Any algebraic number will, of course, be the root of a polynomial with integer coefficients. If number  is a root of a non-zero polynomial , with integer coefficients, then, multiplying this polynomial by . We make sure that  will be the root of the polynomial  with integer coefficients and leading coefficient 1. That's why  integer algebraic number and , that is, any algebraic number is equal to the ratio of an integer algebraic number to an integer rational number.

Let's understand the concept in more detail and compare algebraic numbers with transcendental ones.

Let K be a subfield of E.

Definition: Element  is called algebraic over , if there exists a non-zero polynomial with coefficients in , having  its root.

Definition: Element  called transcendent over , if there is no non-zero polynomial with coefficients in , having  its root.

Let us give examples of transcendental elements. Let  any field and  поле rational fractions from unknowns  over .

Recall that  made up of all sorts of fractions , where  and  polynomials in , , . . . with coefficients in  and .

In this case, the usual definition of the equality of fractions and the usual rules for adding and multiplying fractions take place:

;                                   ;

;

Elements , , . . . field  transcendent over , since they are not roots of non-zero polynomials with coefficients in . Note that any purely transcendental (defined below) extension of the field .

Consider the following definition.

Let the element  algebraic over . Minimum element polynomial  over  is a polynomial of the least degree among nonzero polynomials with coefficients in , having  with its root.

Theorem: Let an element  algebraic over . Let  — minimal element polynomial  over . Then the following statements are true:

1.   irreducible over ;

2. any polynomial with coefficients in , having  its root is divisible by ;

3. any minimal element polynomial  over  has the form , where , a ;

4. any polynomial with coefficients in , irreducible over  and having  its root, will be the minimal polynomial of the element  over .

The main method of the theory of algebraic numbers is the algebra of polynomials. Let's see what means of polynomial algebra work in the proof of this theorem.

As is well known, the degree polynomial is called irreducible over a given field if it cannot be decomposed into a product of polynomials of lesser degree with coefficients in this field. Irreducibility  easiest to prove by contradiction.

Let's assume that , where both factors have coefficients only in  and lesser degree. Substitute into this equality α instead of : . Since the work  field elements , then it follows from here  or . This is contrary to the minimum . Means  irreducible.

Let further  и . Let's apply the division algorithm with a remainder:

,     .

Substitute into this equality  instead of :

Because , that , because minimal polynomial and       . That's why  divided by .

Assertion (3) follows from the fact that two minimal polynomials of the element  over  have equal degree and divide each other, and assertion (4) follows from (2). Thus, in the proof of this theorem, the concept of degree, the division theorem with remainder, and the concept of irreducibility were used.

Also, in the process of studying the topic of algebraic numbers, it is important to understand the concept of the greatest common divisor and the least common multiple.

Polynomial   is called a common divisor for polynomials  и , if it is a divisor for each of these polynomials.

Greatest common divisor of non-zero polynomials  and  such a polynomial is called , which is their common divisor and is divisible by any common divisor of these polynomials. They designate it like this:

.