In
mathematics, a quadratically closed field is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
in which every element has a
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
.
[Lam (2005) p. 33][Rajwade (1993) p. 230]
Examples
* The field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s is quadratically closed; more generally, any
algebraically closed field is quadratically closed.
* The field of
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
is not quadratically closed as it does not contain a square root of −1.
* The union of the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
s
for ''n'' ≥ 0 is quadratically closed but not algebraically closed.
[
* The field of ]constructible number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
s is quadratically closed but not algebraically closed.[Lam (2005) p. 220]
Properties
* A field is quadratically closed if and only if it has universal invariant
In mathematics, the universal invariant or ''u''-invariant of a field describes the structure of quadratic forms over the field.
The universal invariant ''u''(''F'') of a field ''F'' is the largest dimension of an anisotropic quadratic space over ...
equal to 1.
* Every quadratically closed field is a Pythagorean field In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lamb ...
but not conversely (for example, R is Pythagorean); however, every non-formally real
In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions
The definition given above i ...
Pythagorean field is quadratically closed.[
* A field is quadratically closed if and only if its ]Witt–Grothendieck ring
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
Definition
Fix a field ''k'' of characteristic not equal to two. All vector spaces ...
is isomorphic to Z under the dimension mapping.[Lam (2005) p. 34]
* A formally real Euclidean field
In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in .
The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every ...
''E'' is not quadratically closed (as −1 is not a square in ''E'') but the quadratic extension ''E''() is quadratically closed.[
* Let ''E''/''F'' be a finite ]extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
where ''E'' is quadratically closed. Either −1 is a square in ''F'' and ''F'' is quadratically closed, or −1 is not a square in ''F'' and ''F'' is Euclidean. This "going-down theorem" may be deduced from the Diller–Dress theorem In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lamb ...
.[Lam (2005) p.270]
Quadratic closure
A quadratic closure of a field ''F'' is a quadratically closed field containing ''F'' which embeds in any quadratically closed field containing ''F''. A quadratic closure for any given ''F'' may be constructed as a subfield of the algebraic closure
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.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
''F''alg of ''F'', as the union of all iterated quadratic extensions of ''F'' in ''F''alg.[
]
Examples
* The quadratic closure of R is C.[
* The quadratic closure of F5 is the union of the .][
* The quadratic closure of Q is the field of complex ]constructible numbers
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is c ...
.
References
*
* {{cite book , title=Squares , volume=171 , series=London Mathematical Society Lecture Note Series , first=A. R. , last=Rajwade , publisher=Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambr ...
, year=1993 , isbn=0-521-42668-5 , zbl=0785.11022
Field (mathematics)