<abstract xmlns="http://www.w3.org/1999/xhtml">

Let ℙ be the set of prime numbers, ℙ the union ℙ ∪ {0}, and for any field <italic>E</italic>, let char(<italic>E</italic>) be its characteristic, ddim(<italic>E</italic>) the Diophantine dimension of <italic>E</italic>, <italic>𝒢</italic>E the absolute Galois group of <italic>E</italic>, and cd(<italic>𝒢</italic>E) the Galois cohomological dimension <italic>𝒢</italic>E. The research presented in this paper is motivated by the open problem of whether cd(<italic>𝒢</italic>E) ≤ ddim(<italic>E</italic>). It proves the existence of quasifinite fields Φq : <italic>q</italic> ∈ ℙ, with ddim(Φq) infinity and char(Φq) = <italic>q</italic>, for each <italic>q</italic>. It shows that for any integer <italic>m ></italic> 0 and <italic>q ∈</italic> ℙ, there is a quasifinite field Φm,q such that char(Φm,q) = <italic>q</italic> and ddim(Φm,q) = <italic>m</italic>. This is used for proving that for any <italic>q</italic> ∈ ℙ and each pair <italic>k</italic>, <italic>ℓ ∈</italic> (𝕅 ∪ {0, ∞}) satisfying <italic>k</italic> ≤ <italic>ℓ</italic>, there exists a field <italic>E</italic>k,ℓ;q with char(<italic>E</italic>k,ℓ;q) = <italic>q</italic>, ddim(<italic>E</italic>k,ℓ;q) = <italic>ℓ</italic> and cd(<italic>𝒢</italic>Ek,ℓ;q) = <italic>k</italic>. Finally, we show that the field <italic>E</italic>k,ℓ;q can be chosen to be perfect unless <italic>k</italic> = 0 ≠ = <italic>ℓ</italic>.
</abstract>

Let ℙ be the set of prime numbers, ℙ the union ℙ ∪ {0}, and for any field E, let char(E) be its characteristic, ddim(E) the Diophantine dimension of E, 𝒢E the absolute Galois group of E, and cd(𝒢E) the Galois cohomological dimension 𝒢E. The research presented in this paper is motivated by the open problem of whether cd(𝒢E) ≤ ddim(E). It proves the existence of quasifinite fields Φq : q ∈ ℙ, with ddim(Φq) infinity and char(Φq) = q, for each q. It shows that for any integer m > 0 and q ∈ ℙ, there is a quasifinite field Φm,q such that char(Φm,q) = q and ddim(Φm,q) = m. This is used for proving that for any q ∈ ℙ and each pair k, ℓ ∈ (𝕅 ∪ {0, ∞}) satisfying k ≤ ℓ, there exists a field Ek,ℓ;q with char(Ek,ℓ;q) = q, ddim(Ek,ℓ;q) = ℓ and cd(𝒢Ek,ℓ;q) = k. Finally, we show that the field Ek,ℓ;q can be chosen to be perfect unless k = 0 ≠ = ℓ.

Quasifinite fields of prescribed characteristic and Diophantine dimension

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Quasifinite fields of prescribed characteristic and Diophantine dimension"}

Let ℙ be the set of prime numbers, ℙ the union ℙ ∪ {0}, and for any field E, let char(E) be its characteristic, ddim(E) the Diophantine dimension of E, 𝒢E the...

Quasifinite fields of prescribed characteristic and Diophantine dimension

Published Online: Jul 10, 2024

Page range: 19 - 42

Received: May 25, 2023

Accepted: Oct 22, 2023

DOI: https://doi.org/10.2478/auom-2024-0017

Keywords
Quasifinite field, form, field of type, Diophantine dimension, (Galois) cohomological dimension, Laurent (formal power) series field

© 2024 Ivan D. Chipchakov et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Quasifinite fields of prescribed characteristic and Diophantine dimension

Published Online: Jul 10, 2024

Page range: 19 - 42

Received: May 25, 2023

Accepted: Oct 22, 2023

DOI: https://doi.org/10.2478/auom-2024-0017

KeywordsQuasifinite field, form, field of type, Diophantine dimension, (Galois) cohomological dimension, Laurent (formal power) series field

© 2024 Ivan D. Chipchakov et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Quasifinite field, form, field of type, Diophantine dimension, (Galois) cohomological dimension, Laurent (formal power) series field