Sage code for MacLane valuations and applications
Implementations of MacLane's Algorithm for finding all valuations of a finite extension of a field K that extend a valuation v on K, and MacLane's construction of all valuations on the polynomial ring K[x] extending v.
REFERENCES:
.. [MacLane-1] Saunders MacLane, "A Construction for Absolute Values in Polynomial Rings", Trans. AMS., vol. 40 no. 3 (Dec. 1936), pp. 363-395
.. [MacLane-2] Saunders MacLane, "A Construction for Prime Ideals as Absolute Values of an Algebraic Field", Duke Math. J., v.2 (1936), pp. 492-510
.. [Serre] Jean-Pierre Serre, "Corps Locaux", Hermann, Paris (1968).
AUTHORS:
- Xander Faber awfaber@super.org, IDA/Center for Computing Sciences
- David Zelinsky dsz@dedekind.net
TABLE OF CONTENTS
-
Utility Functions
- expand_polynomial
- reduce_at_infinity
- newton_slopes
- polynomial_from_inductive_valuation
- inductive_valuation_from_invariants
-
ExtensionOfFiniteField (class)
-
InductiveValuation (class)
-
StageZeroValuation (constructor)
-
ExtensionFieldValuation (class)
-
ExtensionFieldDecomposition (class)
-
Applications
- p_adic_inductive_valuation
- p_adic_decomposition
- number_field_decomposition
- function_field_inductive_valuation
- function_field_decomposition
-
Construction and Visualization
- compute_vertex_positions
- sort_indvals
- sort_decomp_graph
- decomp_graph
- indvals_from_decomp_graph
- polynomial_from_indvals
- polynomial_from_decomp_graph