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21413 A class for ring extensions caruso "Sage provides a rich framework for dealing with all classical algebraic structures: rings, fields, algebras, etc.
Nevertheless, given (for instance) two fields K and L with K \subset L, it is not possible to build the extension L/K as a Sage object. However one can easily imagine methods related to this extension (e.g. `degree`, `discriminant`, `normal_basis`, `decompose_on_basis`, etc.)
With Bruno Grenet, Johan Rosenkilde and Luca De Feo, we raised this issue at Sage Days 75. A summary of our discussion is available
[https://gist.github.com/defeo/cabab27ea93aeb9e0deb0ba8c5bc745b here].
This ticket implements a generic class for ring extensions, and more specific classes for finite free ring extensions (as finite degree field extensions).
Below is a small tutorial extracted from the documentation:
{{{
Extension of rings.
Sage offers the possibility to work with ring extensions `L/K` as
actual parents and perform meaningful operations on them and their
elements.
The simplest way to build an extension is to use the method
:meth:`sage.rings.ring.CommutativeRing.over` on the top ring,
that is `L`.
For example, the following line constructs the extension of
finite fields `\mathbf{F}_{5^4}/\mathbf{F}_{5^2}`::
sage: GF(5^4).over(GF(5^2))
Field in z4 with defining polynomial x^2 + (4*z2 + 3)*x + z2 over its base
By default, Sage reuses the canonical generator of the top ring
(here `z_4 \in \mathbf{F}_{5^4}`), together with its name. However,
the user can customize them by passing in appropriate arguments::
sage: F = GF(5^2)
sage: k = GF(5^4)
sage: z4 = k.gen()
sage: K. = k.over(F, gen = 1-z4)
sage: K
Field in a with defining polynomial x^2 + z2*x + 4 over its base
The base of the extension is available via the method :meth:`base` (or
equivalently :meth:`base_ring`)::
sage: K.base()
Finite Field in z2 of size 5^2
It also possible to building an extension on top of another extension,
obtaining this way a tower of extensions::
sage: L.** = GF(5^8).over(K)
sage: L
Field in b with defining polynomial x^2 + (4*z2 + 3*a)*x + 1 - a over its base
sage: L.base()
Field in a with defining polynomial x^2 + z2*x + 4 over its base
sage: L.base().base()
Finite Field in z2 of size 5^2
The method :meth:`bases` gives access to the complete list of rings in
a tower::
sage: L.bases()
[Field in b with defining polynomial x^2 + (4*z2 + 3*a)*x + 1 - a over its base,
Field in a with defining polynomial x^2 + z2*x + 4 over its base,
Finite Field in z2 of size 5^2]
Once we have constructed an extension (or a tower of extensions), we
have interesting methods attached to it. As a basic example, one can
compute a basis of the top ring over any base in the tower::
sage: L.basis_over(K)
[1, b]
sage: L.basis_over(F)
[1, a, b, a*b]
When the base is omitted, the default is the natural base of the extension::
sage: L.basis_over()
[1, b]
The method :meth:`sage.rings.ring_extension_element.RingExtensionWithBasis.vector`
computes the coordinates of an element according to the above basis::
sage: u = a + 2*b + 3*a*b
sage: u.vector() # over K
(a, 2 + 3*a)
sage: u.vector(F)
(0, 1, 2, 3)
One can also compute traces and norms with respect to any base of the tower::
sage: u.trace() # over K
(2*z2 + 1) + (2*z2 + 1)*a
sage: u.trace(F)
z2 + 1
sage: u.trace().trace() # over K, then over F
z2 + 1
sage: u.norm() # over K
(z2 + 1) + (4*z2 + 2)*a
sage: u.norm(F)
2*z2 + 2
And minimal polynomials::
sage: u.minpoly()
x^2 + ((3*z2 + 4) + (3*z2 + 4)*a)*x + (z2 + 1) + (4*z2 + 2)*a
sage: u.minpoly(F)
x^4 + (4*z2 + 4)*x^3 + x^2 + (z2 + 1)*x + 2*z2 + 2
}}}" enhancement closed major sage-9.1 algebra fixed sd75, padicBordeaux jsrn defeo bruno nthiery SimonKing saraedum Xavier Caruso David Roe, Frédéric Chapoton N/A c43507cd8a4f320690fa164c46f6f027b24c019a #26105
**