Information about Algebraically Closed Field
In mathematics, a field 
is said to be algebraically closed if every polynomial in one variable of degree at least 
, with coefficients in , has a zero (root) in .
has no solution in real numbers, even though all its coefficients (1, 0 and 1) are real. The same argument proves that the field of rational numbers is not algebraically closed either. Also, no finite field is algebraically closed, because if 
, 
, …, 
are the elements of , then the polynomial
has no zero in. By contrast, the field of complex numbers is algebraically closed: this is stated by the fundamental theorem of algebra. Another example of an algebraically closed field is the field of (complex) algebraic numbers.
, the assertion “ is algebraically closed” is equivalent to each one of the following:
is an algebraically closed field, 
is an element of , and 
is a natural number, then has an th root in , since this is the same thing as saying that the equation 
has some root in . However, there are fields in which every element has an th root (for each natural number ) but which are not algebraically closed. In fact, even assuming that every polynomial of the form 
splits into linear factors is not enough to assure that the field is algebraically closed.
Assuming Zorn's lemma, every field has a unique algebraic closure, which is the smallest algebraically closed field of which is a subfield.
is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in .
Examples
As an example, the field of real numbers is not algebraically closed, because the polynomial equation
has no solution in real numbers, even though all its coefficients (1, 0 and 1) are real. The same argument proves that the field of rational numbers is not algebraically closed either. Also, no finite field
is algebraically closed, because if , , …, are the elements of , then the polynomial
has no zero in
. By contrast, the field of complex numbers is algebraically closed: this is stated by the fundamental theorem of algebra. Another example of an algebraically closed field is the field of (complex) algebraic numbers.
Equivalent properties
Given a field, the assertion “ is algebraically closed” is equivalent to each one of the following:
- Every polynomial
of degree ≥ , with coefficients in , splits into linear factors. In other words, there are elements 
, 
, 
, …, 
of the field such that
- :
- The field has no proper algebraic extension.
- For each natural number , every linear map from
into itself has some eigenvector.
- Every rational function in one variable
, with coefficients in , can be written as the sum of a polynomial function with rational functions of the form 
, where is a natural number, and and 
are elements of .
Other properties
If is an algebraically closed field, is an element of , and is a natural number, then has an th root in , since this is the same thing as saying that the equation has some root in . However, there are fields in which every element has an th root (for each natural number ) but which are not algebraically closed. In fact, even assuming that every polynomial of the form splits into linear factors is not enough to assure that the field is algebraically closed.
Assuming Zorn's lemma, every field
has a unique algebraic closure, which is the smallest algebraically closed field of which is a subfield.
References
- S. Lang, Algebra, Springer-Verlag, 2004, ISBN 0-387-95385-X
- B. L. van der Waerden, Algebra I, Springer-Verlag, 1991, ISBN 0-387-97424-5
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. is a polynomial.
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variable (IPA pronunciation: [ˈvæɹiəbl]) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression.
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coefficient is a constant multiplicative factor of a certain object. For example, the coefficient in 9x2 is 9.
The object can be such things as a variable, a vector, a function, etc.
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The object can be such things as a variable, a vector, a function, etc.
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root (or a zero) of a complex-valued function is a member of the domain of such that vanishes at , that is,
In other words, a "root" of a function is a value for that produces a result of zero ("0").
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In other words, a "root" of a function is a value for that produces a result of zero ("0").
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root (or a zero) of a complex-valued function is a member of the domain of such that vanishes at , that is,
In other words, a "root" of a function is a value for that produces a result of zero ("0").
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In other words, a "root" of a function is a value for that produces a result of zero ("0").
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction , where b is not zero.
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In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory.
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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In mathematics, the fundamental theorem of algebra states that every non-zero single-variable polynomial, with complex coefficients, has exactly as many complex roots as its degree, if repeated roots are counted up to their multiplicity.
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In mathematics, an algebraic number is a number x which satisfies the equality of (and thereby is defined to be a root of) an algebraic equation, i.e. an equation of form a0xn + a1xn-1 + ...
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coefficient is a constant multiplicative factor of a certain object. For example, the coefficient in 9x2 is 9.
The object can be such things as a variable, a vector, a function, etc.
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The object can be such things as a variable, a vector, a function, etc.
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factorization (British English: also factorisation) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original.
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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.
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions.
where P and
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Definitions
In the case of one variable, x, a rational function is a function of the formwhere P and
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Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states:
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Every partially ordered set, in which every chain (i.e. totally ordered subset) has an upper bound, contains at least one maximal element.
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In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics.
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Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was a member of the Bourbaki group.
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Bartel Leendert van der Waerden (February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, Switzerland) was a Dutch mathematician.
Van der Waerden learned advanced mathematics at the University of Amsterdam and the University of Göttingen, from 1919 until 1926.
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Van der Waerden learned advanced mathematics at the University of Amsterdam and the University of Göttingen, from 1919 until 1926.
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