** Every field has an algebraic closure.

In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

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.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

The algebraic closure of a field K can be thought of as the largest algebraic extension of K.

To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K.

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 >.

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.

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.

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.

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 ).

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.

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.

The converse, however, is not true: there are algebraic numbers which cannot be obtained in this manner.

An algebraic integer is an algebraic number which is a root of a polynomial with integer coefficients with leading coefficient 1 ( a monic polynomial ).

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory.

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.

The two values can also be interpreted as logical values ( true / false, yes / no ), algebraic signs (+/−), activation states ( on / off ), or any other two-valued attribute.

The spaceX * of all linear maps into K ( which is called the algebraic dual space to distinguish it from X ′) also induces a weak topology which is finer than that induced by the continuous dual since X ′ ⊆ X *.

C *- algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.

Diophantine geometry, which is the application of techniques from algebraic geometry in this field, has continued to grow as a result ; since treating arbitrary equations is a dead end, attention turns to equations that also have a geometric meaning.

* A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure.

Algebra also defines the rules and conventions of how it is written ( called algebraic notation ).

The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.

Analytically, x can also be raised to an irrational power ( for positive values of x ); the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.

* The fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only ' see ' finite coverings of an algebraic variety.

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.

A related class of numbers are closed-form numbers, which may be defined in various ways, including rational numbers ( and in some definitions all algebraic numbers ), but also allow exponentiation and logarithm.

Aspects of combinatorics include counting the structures of a given kind and size ( enumerative combinatorics ), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ( as in combinatorial designs and matroid theory ), finding " largest ", " smallest ", or " optimal " objects ( extremal combinatorics and combinatorial optimization ), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems ( algebraic combinatorics ).

; Algebraic numbers: The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers.

In mathematics, the nth taxicab number, typically denoted Ta ( n ) or Taxicab ( n ), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways.

Aspects include " counting " the objects satisfying certain criteria ( enumerative combinatorics ), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria ( as in combinatorial designs and matroid theory ), finding " largest ", " smallest ", or " optimal " objects ( extremal combinatorics and combinatorial optimization ), and finding algebraic structures these objects may have ( algebraic combinatorics ).

A narrower definition proposed in, denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm — this need not be algebraically closed, and correspond to explicit algebraic, exponential, and logarithmic operations.