In mathematics, the Pythagoras number or reduced height of 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 ...

describes the structure of the set of squares in the field. The Pythagoras number ''p''(''K'') of a field ''K'' is the smallest positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

''p'' such that every sum of squares in ''K'' is a sum of ''p'' squares. 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 ...

'' is a field with Pythagoras number 1: that is, every sum of squares is already a square.

Examples

* Every non-negative

real number 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 a square, so ''p''(R) = 1. * For a

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, subtr ...

of odd characteristic, not every element is a square, but all are the sum of two squares,Lam (2005) p. 36 so ''p'' = 2. * By

Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four. p = a_0^2 + a_1^2 + a_2^2 + a_ ...

, every positive

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

is a sum of four squares, and not all are sums of three squares, so ''p''(Q) = 4.

Properties

* Every positive integer occurs as the Pythagoras number of some

formally real field 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 ...

.Lam (2005) p. 398 * The Pythagoras number is related to the Stufe by ''p''(''F'') ≤ ''s''(''F'') + 1.Rajwade (1993) p. 44 If ''F'' is not formally real then ''s''(''F'') ≤ ''p''(''F'') ≤ ''s''(''F'') + 1,Rajwade (1993) p. 228 and both cases are possible: for ''F'' = C we have ''s'' = ''p'' = 1, whereas for ''F'' = F₅ we have ''s'' = 1, ''p'' = 2.Rajwade (1993) p. 261 * The Pythagoras number is related to the

height of a field 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 space ...

''F'': if ''F'' is formally real then ''h''(''F'') is the smallest power of 2 which is not less than ''p''(''F''); if ''F'' is not formally real then ''h''(''F'') = 2''s''(''F'').Lam (2005) p. 395 As a consequence, the Pythagoras number of a non-formally-real field, if finite, is either a power of 2 or 1 less than a power of 2, and all cases occur.Lam (2005) p. 396

Notes

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 King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...

, year=1993 , isbn=0-521-42668-5 , zbl=0785.11022 Field (mathematics) Sumsets