# 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