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  2. Algebraic number - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number

    An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (+) /, is an algebraic number, because it is a root of the polynomial x 2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero.

  3. List of types of numbers - Wikipedia

    en.wikipedia.org/wiki/List_of_types_of_numbers

    Algebraic number: Any number that is the root of a non-zero polynomial with rational coefficients. Transcendental number: Any real or complex number that is not algebraic. Examples include e and π. Trigonometric number: Any number that is the sine or cosine of a rational multiple of π.

  4. Algebraic number theory - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number_theory

    Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields .

  5. Number theory - Wikipedia

    en.wikipedia.org/wiki/Number_theory

    An algebraic number is any complex number that is a solution to some polynomial equation () = with rational coefficients; for example, every solution of + (/) + = (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields.

  6. Algebraic number field - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number_field

    The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods.

  7. Algebraic integer - Wikipedia

    en.wikipedia.org/wiki/Algebraic_integer

    If x is an algebraic number then a n x is an algebraic integer, where x satisfies a polynomial p(x) with integer coefficients and where a n x n is the highest-degree term of p(x). The value y = a n x is an algebraic integer because it is a root of q(y) = a n − 1 n p(y /a n), where q(y) is a monic polynomial with integer coefficients.

  8. Field (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Field_(mathematics)

    For example, the algebraic closure Q of Q is called the field of algebraic numbers. The field F is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice. [34] In this regard, the algebraic closure of F q, is exceptionally simple.

  9. Real closed field - Wikipedia

    en.wikipedia.org/wiki/Real_closed_field

    For example, the real closure of the ordered field of rational numbers is the field of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier , who proved it in 1926. If ( F , P ) is an ordered field, and E is a Galois extension of F , then by Zorn's lemma there is a maximal ordered field extension ( M , Q ) with M a ...

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