### Abstract

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_{n} over k(x) such that [K : k(x)] = p^{n} using the concept of Witt vectors. This is accomplished in the following way; if [β_{1}, β_{2},..., β_{n}] is a Witt vector over k(x) = K_{0}, then the Witt equation y^{p} • y = β generates a tower of extensions through K_{i} = K_{i-1}(y_{i}) where y = [y_{1}, y_{2},..., y_{n}]. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_{n}. This alternate generation has the form K_{i} = K_{i-1}(y_{i}); y_{i}^{p} - y_{i} = B_{i}, where, as a divisor in K_{i-1}, B_{i} has the form (B_{i}) = q Πp_{j}^{λj}. In this form q is prime to Πp_{j}^{λj} and each λ_{j} is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_{n} over an algebraically closed field of constants.

Original language | English (US) |
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Pages (from-to) | 303-323 |

Number of pages | 21 |

Journal | Journal of Number Theory |

Volume | 10 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1978 |

### ASJC Scopus subject areas

- Algebra and Number Theory