Unramified degree n extension of Q_p

For a prime $p$ and $n\ge1$ , $\mathbb Q_p^{(n),\mathrm{ur}}$ denotes an unramified extension of $\mathbb Q_p$ of degree $n$ , equipped with the extended $p$ -adic topology. Its ramification index is one and its residue field is $\mathbb F_{p^n}$ .

Choice and notation

Within a chosen algebraic closure of $\mathbb Q_p$ , there is a unique such subextension of each positive degree. The displayed notation is local catalogue notation; the algebraic closure and embedding are parameters when needed. Its valuation ring has uniformizer $p$ .

Degree-one boundary

Degree one recovers $\mathbb Q_p$ . Higher-degree examples give characteristic-zero local fields with the same residue-field sizes as the equal-characteristic Laurent-series examples; these fields are not isomorphic because their characteristics differ.

References

J. S. Milne, Algebraic Number Theory, Chapter 7, Proposition 7.50, unramified extensions, pp. 127–128.