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A263800

Numbers n such that there is some field with u-invariant n.
(history; published version)

#9 by N. J. A. Sloane at Fri Dec 18 11:38:52 EST 2015

STATUS

editing

approved

#8 by N. J. A. Sloane at Fri Dec 18 11:38:48 EST 2015

KEYWORD

hard,nonn,changed,more

STATUS

proposed

editing

#7 by Michel Marcus at Fri Dec 18 10:51:06 EST 2015

STATUS

editing

proposed

Discussion

Fri Dec 18

10:55

Omar E. Pol: Keyword "more" ?

#6 by Michel Marcus at Fri Dec 18 10:50:55 EST 2015

REFERENCES

Oleg T. Izhboldin, Fields of u-Invariant 9, Annals of Mathematics Second Series, 154:3 (Nov 2001), pp. 529-587.

LINKS

Oleg T. Izhboldin, <a href="http://www.jstor.org/stable/3062141">Fields of u-Invariant 9</a>, Annals of Mathematics, Second Series, 154:3 (Nov 2001), pp. 529-587.

STATUS

proposed

editing

#5 by Charles R Greathouse IV at Fri Dec 18 10:04:35 EST 2015

STATUS

editing

proposed

#4 by Charles R Greathouse IV at Mon Oct 26 19:56:55 EDT 2015

LINKS

Wikipedia, <a href="https://en.wikipedia.org/wiki/Isotropic_quadratic_form">Isotropic quadratic form</a>

Wikipedia, <a href="https://en.wikipedia.org/wiki/U-invariant">u-invariant</a>

Discussion

Thu Dec 17

07:23

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#3 by Charles R Greathouse IV at Mon Oct 26 19:09:01 EDT 2015

COMMENTS

It is folklore that 3, 5, and 7 are not in this sequence, see for example Proposition 6.8 in Lam 2005 chapter XI. Merkurjev showed that a(4) = 6, and more generally (unpublishedsee Merkurʹev 1991) that 2n is in this sequence for n > 0. Izhboldin showed that a(6) = 9. It is not known if 11, 13, 15, ... are in this sequence, see Question 6.4 in chapter XIII in Lam 2005.

REFERENCES

A. S. Merkurʹev, Simple algebras and quadratic forms (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 55:1 (1991), pp. 218-224; translation in Math. USSR-Izv. 38:1 (1992), pp. 215-221.

#2 by Charles R Greathouse IV at Mon Oct 26 19:02:51 EDT 2015

NAME

allocated for Charles R Greathouse IVNumbers n such that there is some field with u-invariant n.

DATA

1, 2, 4, 6, 8, 9, 10

OFFSET

1,2

COMMENTS

Let F be a field of characteristic other than 2. Call a quadratic form over F isotropic if it represents zero nontrivially over F, or anisotropic otherwise. The u-invariant of a field F is the supremum of the dimensions of anisotropic quadratic forms over F.

It is folklore that 3, 5, and 7 are not in this sequence, see for example Proposition 6.8 in Lam 2005 chapter XI. Merkurjev showed that a(4) = 6, and more generally (unpublished) that 2n is in this sequence for n > 0. Izhboldin showed that a(6) = 9. It is not known if 11, 13, 15, ... are in this sequence, see Question 6.4 in chapter XIII in Lam 2005.

REFERENCES

Oleg T. Izhboldin, Fields of u-Invariant 9, Annals of Mathematics Second Series, 154:3 (Nov 2001), pp. 529-587.

Tsit-Yuen Lam, Fields of u-invariant 6 after A. Merkurjev, Ring theory 1989. In honor of S. A. Amitsur, Proc. Symp. and Workshop, Jerusalem/Isr. 1988/89, Isr. Math. Conf. Proc. 1, 12-30 (1989).

Tsit-Yuen Lam, Introduction to Quadratic Forms Over Fields (2005); 550 pp.

A. S. Merkurjev, Simple algebras over function fields of quadrics, manuscript (1989), 6 pp.

EXAMPLE

u(F) = 1 for all quadratically-closed fields (like C), so 1 is in the sequence.

u(F) = 2 for all finite fields (like F_2), so 2 is in the sequence.

u(F) is not 3 for any field F, so 3 is not in the sequence.

KEYWORD

allocated

hard,nonn

AUTHOR

Charles R Greathouse IV, Oct 26 2015

STATUS

approved

editing

#1 by Charles R Greathouse IV at Mon Oct 26 19:02:51 EDT 2015

NAME

allocated for Charles R Greathouse IV

KEYWORD

allocated

STATUS

approved