Rings and their Fields

This section is concerned with fields (mainly local and global arithmetic fields), their rings of integers and valuation rings.

• The rational field $\mbox{\bf Q}$ and its ring of integers $\mbox{\bf Z}$
• Residue class rings of $\mbox{\bf Z}$
• Univariate polynomial rings
• Finite fields
• Number fields and their orders
• Rational function fields
• Algebraic function fields
• Valuation rings
• Real and complex fields
• Local fields
• Power series rings and Laurent series rings

In the case of arithmetic fields, the major facilities include:

• Construction of a basis for the maximal order(s)
• Decomposition of ideals into prime ideals
• Recognition of principal ideals
• The class group
• Fundamental units