Let there be a ring r of polynomials in six indeterminates t,u,v,w,x,y with complex numbers as coefficients. Take then a quadratic extension (if this is the right word) R of r by adding a new letter z which is the square root of a polynomial p in our six indeterminates (these latter are, of course, transcendental over C). Polynomial p is homogeneous of 14th degree, if this matters. The question is: where can I read about (algorithms of) factorizing elements in R? Answer for only homogeneous elements will suffice (assuming z has degree 7), but if a factorization is not unique, I want all of them!
Remark. Straightforward attempts using primary decomposition algorithm in Singular proved to be beyond the capabilities of my computer.
Here is an article that discusses the type of factorization you mention. (For what it's worth, I regard this as a very good article in this particular area.)
Mahdi Javadi and Michael Monagan.
On Factorization of Multivariate Polynomials over Algebraic Number and Function Fields.
Proceedings of ISSAC '09, 199–206, ACM Press, 2009.
http://www.cecm.sfu.ca/personal/monaganm/papers/mahdiFactor.pdf
https://www.researchgate.net/publication/221564800_On_factorization_of_multivariate_polynomials_over_algebraic_number_and_function_fields?ev=prf_pub
Conference Paper On factorization of multivariate polynomials over algebraic ...
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