Fields of u-invariant 11

The u-invariant of a field is the highest dimension of a non-degenerate anisotropic quadratic form over this field. As known since the 50es, the set of possible finite values of the u-invariant starts with 1, excludes 3, 5, 7, and includes all 2-powers. It was shown by Alexander Merkurjev in the end of the 80es that this set contains 6 and -- a couple of years later -- all positive even integers. Oleg Izhboldin proved by the end of the 90es that 9 is also there. In the second half of the 00s, this result has been extended to all larger numbers of the form a 2-power plus 1 by Alexander Vishik. Here we show that the value 11 is taken. The result still holds if we restrict to fields of any fixed characteristic.