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

Let | | be a discrete non-archimedean absolute value of a field <italic>K</italic> with valuation ring 𝒪, maximal ideal <italic>𝓜</italic> and residue field 𝔽 = <italic>𝒪/𝓜</italic>. Let <italic>L</italic> be a simple finite extension of <italic>K</italic> generated by a root <italic>α</italic> of a monic irreducible polynomial <italic>F</italic> ∈ <italic>O</italic>[<italic>x</italic>]. Assume that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2023-0007_ieqn_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>F</mi></mrow><mrow><mo stretchy="true">¯</mo></mrow></mover><mo>=</mo><msup><mrow><mover accent="true"><mrow><mi>ϕ</mi></mrow><mrow><mo stretchy="true">¯</mo></mrow></mover></mrow><mrow><mi>l</mi></mrow></msup></mrow></math><tex-math>$\overline F = \overline \varphi ^l$</tex-math></alternatives></inline-formula> in 𝔽[<italic>x</italic>] for some monic polynomial <italic>φ</italic> ∈ <italic>O</italic>[<italic>x</italic>] whose reduction modulo <italic>𝓜</italic> is irreducible, the <italic>φ</italic>-Newton polygon <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2023-0007_ieqn_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><mover accent="true"><mrow><mi>φ</mi></mrow><mrow><mo stretchy="true">¯</mo></mrow></mover><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math><tex-math>$N\overline \phi \left( F \right)$</tex-math></alternatives></inline-formula> has a single side of negative slope λ, and the residual polynomial <italic>R</italic>λ(<italic>F</italic> )(<italic>y</italic>) has no multiple factors in 𝔽<italic>φ</italic>[<italic>y</italic>]. In this paper, we describe all absolute values of <italic>L</italic> extending | |. The problem is classical but our approach uses new ideas. Some useful remarks and computational examples are given to highlight some improvements due to our results.
</abstract>

Let | | be a discrete non-archimedean absolute value of a field K with valuation ring 𝒪, maximal ideal 𝓜 and residue field 𝔽 = 𝒪/𝓜. Let L be a simple finite extension of K generated by a root α of a monic irreducible polynomial F ∈ O[x]. Assume that F¯=ϕ¯l$\overline F  = \overline \varphi  ^l$ in 𝔽[x] for some monic polynomial φ ∈ O[x] whose reduction modulo 𝓜 is irreducible, the φ-Newton polygon Nφ¯(F)$N\overline \phi  \left( F \right)$ has a single side of negative slope λ, and the residual polynomial Rλ(F )(y) has no multiple factors in 𝔽φ[y]. In this paper, we describe all absolute values of L extending | |. The problem is classical but our approach uses new ideas. Some useful remarks and computational examples are given to highlight some improvements due to our results.

On The Geometric Determination of Extensions of Non-Archimedean Absolute Values

Tatra Mountains Mathematical Publications

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

{"article-title":"On The Geometric Determination of Extensions of Non-Archimedean Absolute Values"}

Let | | be a discrete non-archimedean absolute value of a field K with valuation ring 𝒪, maximal ideal 𝓜 and residue field 𝔽 = 𝒪/𝓜. Let L be a simple...

On The Geometric Determination of Extensions of Non-Archimedean Absolute Values

Published Online: Mar 07, 2023

Page range: 87 - 102

Received: Nov 11, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0007

Keywords
extensions of non-archimedean absolute value, Newton polygon, residual polynomial

© 2023 Mathematical Institute, Slovak Academy of Sciences

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

On The Geometric Determination of Extensions of Non-Archimedean Absolute Values

Published Online: Mar 07, 2023

Page range: 87 - 102

Received: Nov 11, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0007

Keywordsextensions of non-archimedean absolute value, Newton polygon, residual polynomial

© 2023 Mathematical Institute, Slovak Academy of Sciences

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

Keywords
extensions of non-archimedean absolute value, Newton polygon, residual polynomial