Trac #29716: Splitting Algebras

''Splitting algebras'' have been considered by [http://www.iumj.indiana.
edu/IUMJ/fulltext.php?artid=4791&year=2012&volume=61 Dan Laksov, Anders
Thorup],
[https://www.worldscientific.com/doi/abs/10.1142/S0219498805001034
Torsten Ekedahl] and others in order to study intersection theory of
Grassmann and other flag schemes. Similarily as ''splitting fields''
they can be considered as extensions of rings containing all the roots
of a given monic polynomial over that ring under the assumption that its
Galois group is the symmetric group of order equal to the polynomial's
degree.

Thus they can be used as a tool to express elements of a ring generated
by `n` indeterminates in terms of symmetric functions in these
indeterminates.

This ticket realizes splitting algebras by a recursive quotient ring
construction splitting off some linear factor of the polynomial in each
recursive step. Accordingly it will be inherited from class
`PolynomialQuotientRing`.

URL: https://trac.sagemath.org/29716
Reported by: soehms
Ticket author(s): Sebastian Oehms
Reviewer(s): Travis Scrimshaw

src/doc/en/reference/algebras/index.rst

@@ -88,6 +88,7 @@ Various associative algebras
    sage/algebras/cellular_basis
    sage/algebras/commutative_dga
    sage/algebras/q_system
+   sage/algebras/splitting_algebra
 
 Non-associative algebras
 ------------------------

src/doc/en/reference/references/index.rst

@@ -1995,6 +1995,9 @@ REFERENCES:
             Version
             1.3". 2013. http://www.wolfgang-ehrhardt.de/specialfunctions.pdf.
 
+.. [EL2002] Ekedahl, Torsten & Laksov, Dan. (2002). *Splitting algebras, Symmetric functions and
+            Galois Theory*. J. Algebra Appl. 4, :doi:`10.1142/S0219498805001034`
+
 .. [EM2001] Pavel Etingof and Xiaoguang Ma.
             *Lecture notes on Cherednik algebras*.
             http://www-math.mit.edu/~etingof/73509.pdf :arxiv:`1001.0432`.
@@ -3425,6 +3428,9 @@ REFERENCES:
              Mémoire de maîtrise en Mathématiques, Montréal, UQAM,
              2008, 109 pages.
 
+.. [Lak2010] Dan Laksov. *Splitting algebras and Gysin homomorphisms*.
+             Journal of Commutative Algebra, Volume 2, Number 3, Fall 2010
+
 .. [Lam1996] \T. K. Lam. *B and D analogues of stable Schubert polynomials and
              related insertion algorithms*. PhD Thesis, MIT, 1996.
 
@@ -3708,6 +3714,9 @@ REFERENCES:
             groups*. Australian Mathematical Society Lecture
             Series, 2009.
 
+.. [LT2012] Dan Laksov, Anders Thorup. *Splitting algebras and Schubert calculus*,
+            Indiana Univ. Math. J. 61 (2012), 1253-1312 :doi:`10.1512/iumj.2012.61.4791`
+
 .. [DeLuca2006] \A. De Luca, *Pseudopalindrome closure operators in free
               monoids*, Theoret. Comput. Sci. 362 (2006) 282--300.
 
@@ -5209,6 +5218,9 @@ REFERENCES:
              height on elliptic curves over number fields, Math. Comp. 79
              (2010), pages 2431-2449.
 
+.. [Tho2011] Anders Thorup. *ON THE INVARIANTS OF THE SPLITTING ALGEBRA*,
+             2011, :arxiv:`1105.4478`
+
 .. [TIDES] \A. Abad, R. Barrio, F. Blesa, M. Rodriguez. TIDES tutorial:
            Integrating ODEs by using the Taylor Series Method
            (http://www.unizar.es/acz/05Publicaciones/Monografias/MonografiasPublicadas/Monografia36/IndMonogr36.htm)