Algebra & Number Theory

On a conjecture of Kato and Kuzumaki

Diego Izquierdo

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In 1986, Kato and Kuzumaki stated several conjectures in order to give a diophantine characterization of cohomological dimension of fields in terms of projective hypersurfaces of small degree and Milnor K-theory. We establish these conjectures for finite extensions of (x1,,xn) and (x1,,xn)((t)), and we prove new local-global principles over number fields and global fields of positive characteristic in the context of Kato and Kuzumaki’s conjectures.

Article information

Algebra Number Theory, Volume 12, Number 2 (2018), 429-454.

Received: 8 June 2017
Revised: 23 October 2017
Accepted: 18 December 2017
First available in Project Euclid: 23 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E76: Forms of degree higher than two
Secondary: 12E25: Hilbertian fields; Hilbert's irreducibility theorem 12E30: Field arithmetic 14G27: Other nonalgebraically closed ground fields 19D45: Higher symbols, Milnor $K$-theory 19F99: None of the above, but in this section

Cohomological dimension of fields $C_i$ property Milnor K-theory Number fields Function fields


Izquierdo, Diego. On a conjecture of Kato and Kuzumaki. Algebra Number Theory 12 (2018), no. 2, 429--454. doi:10.2140/ant.2018.12.429.

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