Local Rings: Construction

• Construction of a p-adic ring or field
• Unramified extension of a local ring or field
• Totally ramified extension of a local ring or field
• Ring of integers of a local field
• Field of fractions of a local ring
• Change precision of a ring, field or element

A local ring is a finite degree extension of a p-adic ring and may be either ramified or unramified or both. It is constructed in the form of a two-step extension, the first step being an unramified extension I of degree f over $\mbox{\bf Z}_p$, generated by a root of unity of order p^f-1, and the second being a totally ramified extension of I defined by an Eisenstein polynomial.