Some versions of the U-invariant of a field

Abstract

Let F be a field, char F ≠ 2 . Assume that a 1 , … a n ∈ F ⁎ are such that a ‾ 1 , … a ‾ n ∈ F ⁎ / F ⁎ 2 are linearly independent over Z / 2 Z . As usual W ( F ) stands for the Witt ring of F . For an element φ ∈ W ( F ) denote by dim φ the dimension of the corresponding anisotropic quadratic form . Define u ˆ ( F ; a 1 , … , a n ) as the maximum of dim φ , where φ runs over the set of elements in W ( F ) , which become zero in W ( F ( a 1 , … , a n ) ) . This is a version of the classical notion of the u -invariant u ( F ) of the field F . It turns out that u ˆ ( F ; a 1 , … , a n ) ≤ α n ∑ i = 1 n u ˆ ( F ; a i ) for any n ≥ 2 , where the sequence α n is defined recurrently as α 2 = 1 , and α n = 5 2 ( n − 1 ) α n − 1 + 1 n . We compute u ˆ ( F ; a ) in certain cases, and show that u ˆ ( F ( b ) ; a ) ≤ 5 2 u ˆ ( F ; a ) , where b ∈ F ⁎ . However, in general there is no lower bound for u ˆ ( F ( b ) ; a ) via u ˆ ( F ; a ) , even though we prove that max ⁡ { u ˆ ( F ( b ) ; a ) , u ˆ ( F ( a b ) ; a ) } ≥ 1 3 u ˆ ( F ; a ) . Let u ˆ ( F ) be the maximum of u ˆ ( F ; a ) , where a runs over all elements of F ⁎ ∖ F ⁎ 2 . We show that u ˆ ( F ( b ) ) ≥ 1 4 u ˆ ( F ) if b is a sum of two squares. In particular, the last inequality holds if − 1 ∈ F ⁎ .