Difference between revisions of "Pythagorean field"
From Encyclopedia of Mathematics
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A [[field]] in which any sum of two squares is a square. The '''Pythagorean closure''' of a field $K$ is the minimal Pythagorean subfield $K^\pi$ of the algebraic closure $\bar K$ containing $K$. | A [[field]] in which any sum of two squares is a square. The '''Pythagorean closure''' of a field $K$ is the minimal Pythagorean subfield $K^\pi$ of the algebraic closure $\bar K$ containing $K$. | ||
| − | The '''Pythagoras number''' of a field $K$ is the smallest integer $d$, if it exists, such that every sum of squares in $K$ is already a sum of at most $d$ squares (or $\infty$ if no such number exists). A Pythagorean field has Pythagoras number equal to $1$. [[Lagrange theorem|Lagrange's theorem]] | + | The '''Pythagoras number''' of a field $K$ is the smallest integer $d$, if it exists, such that every sum of squares in $K$ is already a sum of at most $d$ squares (or $\infty$ if no such number exists). A Pythagorean field has Pythagoras number equal to $1$. [[Lagrange theorem|Lagrange's theorem]] implies that the Pythagoras number of the field of rational numbers is $4$. A finite field has Pythagoras number $1$ (in characteristic $2$) or $2$ (in characteristic $>2$). Every positive integer occurs as the Pythagoras number of some [[formally real field]]. |
====References==== | ====References==== | ||
| − | * Tsit Yuen Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', American Mathematical Society (2005) ISBN 0-8218-1095-2 {{ZBL|1068.11023}} {{MR|2104929 }} | + | * Tsit Yuen Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', American Mathematical Society (2005) {{ISBN|0-8218-1095-2}} {{ZBL|1068.11023}} {{MR|2104929 }} |
| − | * A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) ISBN 0-521-42668-5 {{ZBL|0785.11022}} | + | * A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}} {{ZBL|0785.11022}} |
| − | * J.W. Milnor, D. Husemöller, ''Symmetric bilinear forms'', Ergebnisse der Mathematik und ihrer Grenzgebiete '''73''', Springer-Verlag (1973) ISBN 0-387-06009-X {{ZBL|0292.10016}} | + | * J.W. Milnor, D. Husemöller, ''Symmetric bilinear forms'', Ergebnisse der Mathematik und ihrer Grenzgebiete '''73''', Springer-Verlag (1973) {{ISBN|0-387-06009-X}} {{ZBL|0292.10016}} |
Latest revision as of 19:33, 15 November 2023
A field in which any sum of two squares is a square. The Pythagorean closure of a field $K$ is the minimal Pythagorean subfield $K^\pi$ of the algebraic closure $\bar K$ containing $K$.
The Pythagoras number of a field $K$ is the smallest integer $d$, if it exists, such that every sum of squares in $K$ is already a sum of at most $d$ squares (or $\infty$ if no such number exists). A Pythagorean field has Pythagoras number equal to $1$. Lagrange's theorem implies that the Pythagoras number of the field of rational numbers is $4$. A finite field has Pythagoras number $1$ (in characteristic $2$) or $2$ (in characteristic $>2$). Every positive integer occurs as the Pythagoras number of some formally real field.
References
- Tsit Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929
- A. R. Rajwade, Squares, London Mathematical Society Lecture Note Series 171 Cambridge University Press (1993) ISBN 0-521-42668-5 Zbl 0785.11022
- J.W. Milnor, D. Husemöller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag (1973) ISBN 0-387-06009-X Zbl 0292.10016
How to Cite This Entry:
Pythagorean field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pythagorean_field&oldid=39951
Pythagorean field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pythagorean_field&oldid=39951