# RefReq: Algorithms for standard operations in Algebraic Number theory

Given an algebraic number field $F$ (I actually don't have an idea how to implement this data already, except for splitting fields of polynomials, but there is something in SAGE) is there free code available n the internet for the following operations:

• Compute the ring of integers $O$ and its units
• Factorize a given prime ideal of $\mathbb{Q}$ in $F$ with a choice for generators, ramification degree, etc.
• Compute the Galois groups over $\mathbb{Q}$
• Compute monic, irreducible, quadratic polynomials $X^2 + A X + D$ for $A, D \leq N$ (appropiately interpreted in terms of ideals) and give information at which places it factorizes.

What is the analogue for global function fields?

• Do you know Cohen's book on comptuational algebraic number theory? – Cam McLeman Sep 3 '13 at 13:10
• No, but it seems to cover most of what I want (at least the characteristic zero stuff). At least the available excerpts from Amazon suggest this. I will borrow it from our library now;) – Marc Palm Sep 3 '13 at 13:16
• This book seem to be more about the theoretical background. I am interested more down-to-earth, say, ready-to-use codes in C++, Java, Pari or Sage, even pseudo code. I will try to clarify this. – Marc Palm Sep 3 '13 at 13:37
• Pari/GP has some of this functionality. Type ?6 at the prompt to see what's available. Pari is also part of sage, so will be there also. – Felipe Voloch Sep 3 '13 at 14:00
• Sage and Pari will both do the first three things on your list. I can't quite work out what the fourth one means exactly (but the two programs above can both happily factor polynomials over localizations of number fields). – David Loeffler Sep 3 '13 at 15:54

## 1 Answer

I think the most comprehensive implementations (if not necessarily the fastest, depending on the operation needed) are found in Magma. Magma certainly has functions for function fields as well as number fields.