//compute the isomorphism classes attained by Q3[t] / (t^3 - s(t-2)) for s in Z3 time E := EtaleAlgebraFamily(F, 3);

printf "Found %o isomorphism classes\n", #E; printf "The 12-th one is: %o\n", SimplifyToProduct(E[12]); ```

This produces the following output.

Found 14 isomorphism classes The 12-th one is: Etale algebra defined by product of [ Unramified extension defined by the polynomial x + 2 + O(3^50) over Unramified extension defined by a map over 3-adic field mod 3^50, Totally ramified extension defined by the polynomial x^2 - 3 + O(3^50) over Unramified extension defined by a map over 3-adic field mod 3^50 ] with stable neighbourhoods [ 27 + 1594323 * (OK^2 - {0}), 54 + 243 * OK, 2214 + 6561 * OK, 19710 + 59049 * OK ]

This output above shows that $\mathbb Q3[t] / (t^3 - s0(t-2))$ attains $14$ distinct isomorphism classes for $s0\in\mathbb Z3$ (for which $t^3 - s0(t-2)$ is also separable). For instance, we have $$\mathbb Q3[t] / (t^3 - s0(t-2)) \cong \mathbb Q3(\sqrt{3}) \times \mathbb Q3$$ with $s0\in\mathbb Z3$ if and only if $s0\in 2\cdot 3^3 + 3^5\mathbb Z3$ or $`s0\in 3^3 + 3^7(\mathbb Z_3^2\setminus {0})`$.

More examples can be found in examples.

Contents

We give a quick description of the contents of this Magma package.

• EtaleAlgebras: This folder contains the main functionality for working with étale algebras and polynomials over local fields. We give a quick description for every file.
  • etale_algebra.m: Implementation of étale algebras over a local field.
  • etalealgebrafamily.m: Implements the main function EtaleAlgebraFamily, which can compute isomorphism classes of étale algebras parametrized by a one-variable family
  • padic_nbhd.m: Implements $p$-adic parameter spaces.
  • separant.m: Computing separants of polynomials over local fields, and other related expressions involving the roots of a polynomial over a local field.
  • tschirnhaus.m: Computing Tschirnhaus transformations of defining polynomials of an étale algebra over a local field.
  • utils.m: Contains some miscellaneous functions.
• GFE: This folder contains functions for computing étale algebras arising from Belyi maps and generalized Fermat equations of various signatures. See [1] for
  • 257.m: Functions for the degree $8$ Belyi map corresponding to the generalized Fermat equation of signature $(2,5,7)$ (or permutations thereof).
  • 257_relative.m: Functions for the degree $7$ Belyi map (over $\mathbb{Q} {(}\sqrt{21}{)}$) corresponding to the generalized Fermat equation of signature $(2,5,7)$ (or permutations thereof).
  • 3511.m: Functions for the degree $11$ Belyi map corresponding to the generalized Fermat equation of signature $(3,5,11)$ (or permutations thereof).
  • 357.m: Functions for the degree $7$ Belyi map corresponding to the generalized Fermat equation of signature $(3,5,7)$ (or permutations thereof).
• LocalFields: Contains some useful functionality for finite extensions of $p$-adic fields.
  • localfielddatabase.m: Implements a data structure for pre-computed databases of $p$-adic fields. In particular, it contains LMFDB data for degree $8$ extensions of $\mathbb Q2$ and degree $10$ extensions of $\mathbb Q5$.
  • subfields.m: Functions for computing the subfields of a $p$-adic field.
• examples: This folder contains some code examples using this package.
• scripts: This folder contains various Magma scripts used for the computations in [1].

References

[1] Casper Putz. `Enumeration of local and global étale algebras applied to generalized Fermat equations`. PhD thesis. Vrije universiteit Amsterdam, 2024. https://research.vu.nl/en/publications/enumeration-of-local-and-global-étale-algebras-applied-to-general