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An algebraic integer is an algebraic number which is a root of a polynomial with integer coefficients with leading coefficient 1 ( a monic polynomial ).
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In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).
* The quadratic surds ( irrational roots of a quadratic polynomial with integer coefficients,, and ) are algebraic numbers.
All numbers which can be obtained from the integers using a finite number of integer additions, subtractions, multiplications, divisions, and taking nth roots ( where n is a positive integer ) are algebraic.
Algebraic numbers coloured by leading coefficient ( red signifies 1 for an algebraic integer ).
The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.
* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).
A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization we have the isomorphism between a ring and the direct product of its prime power parts:
In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.
Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable.
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial ( a polynomial whose leading coefficient is 1 ) with coefficients in ( the set of integers ).
Each algebraic integer belongs to the ring of integers of some number field.
A number x is an algebraic integer if and only if the ring is finitely generated as an abelian group, which is to say, as a-module.
The following are equivalent definitions of an algebraic integer.
* is an algebraic integer if there exists a monic polynomial such that.
* is an algebraic integer if the minimal monic polynomial of over is in.
* is an algebraic integer if is a finitely generated-module.
* is an algebraic integer if there exists a finitely generated-submodule such that.
algebraic and is
It is also commonly used in mathematics in algebraic solutions representing quantities such as angles.
Another example of an algebraically closed field is the field of ( complex ) algebraic numbers.
This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers.
** The golden ratio is algebraic since it is a root of the polynomial.
* The set of algebraic numbers is countable ( enumerable ).
* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.
If its minimal polynomial has degree, then the algebraic number is said to be of degree.
An algebraic number of degree 1 is a rational number.
* The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.
The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.
Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.
This can be rephrased by saying that the field of algebraic numbers is algebraically closed.
In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
The converse, however, is not true: there are algebraic numbers which cannot be obtained in this manner.
algebraic and number
* Any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number.
If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.
He is especially known for his foundational work in number theory and algebraic geometry.
He made substantial contributions in many areas, the most important being his discovery of profound connections between algebraic geometry and number theory.
His construction of new cohomology theories has left deep consequences for algebraic number theory, algebraic topology, and representation theory.
Alexander Grothendieck's work during the ` Golden Age ' period at IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis.
In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.
If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.
The term abstract data type can also be regarded as a generalised approach of a number of algebraic structures, such as lattices, groups, and rings.
algebraic and which
The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.
Plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals, are some of the most studied classes of algebraic varieties.
He went on to plan and execute a major foundational programme for rebuilding the foundations of algebraic geometry, which were then in a state of flux and under discussion in Claude Chevalley's seminar ; he outlined his programme in his talk at the 1958 International Congress of Mathematicians.
In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.
The converse is not true however: there are infinite extensions which are algebraic.
Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.
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