Level (algebra)

The level is a term from the mathematical subfield of algebra or its subfield of body theory . It indicates the number of square numbers that you have to add up to be able to represent the number . ${\ displaystyle -1}$

definition

Be a body . Can be represented as the sum of squares in , then called the smallest natural number such that the sum of squares is the level of . If it cannot be represented as a sum of squares, then you bet . ${\ displaystyle F}$ ${\ displaystyle -1}$ ${\ displaystyle F}$ ${\ displaystyle s}$ ${\ displaystyle -1}$ ${\ displaystyle s}$ ${\ displaystyle F}$ ${\ displaystyle -1}$ ${\ displaystyle s = \ infty}$

properties

If it is a non-real body , then the level of is finite. ${\ displaystyle F}$ ${\ displaystyle F}$

For a non-real body, one holds for a . ${\ displaystyle F}$ ${\ displaystyle s (F) = 2 ^ {k}}$ ${\ displaystyle k \ in \ mathbb {N}}$

It applies to all bodies with positive characteristics . ${\ displaystyle s (F) \ leq 2}$ ${\ displaystyle F}$

Individual evidence

↑ John Milnor, Dale Husemoller: Symmetric Bilinear Forms . Springer Berlin Heidelberg, Berlin, Heidelberg 1973, ISBN 978-3-642-88332-3 , pp. 75 .

[Milnor75-1] John Milnor, Dale Husemoller: Symmetric Bilinear Forms . Springer Berlin Heidelberg, Berlin, Heidelberg 1973, ISBN 978-3-642-88332-3 , pp. 75 .