Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$K=k((t))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a local field of characteristic <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with perfect residue field k. Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\vec {a}=(a_0,a_1,\dots ,a_{n-1})\in W_n(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>a</mi> <mo stretchy="false">→</mo> </mover> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi>W</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be a Witt vector of length n. Assume that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\vec {a}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> is “reduced”, and that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$v_K(a_0)<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi>K</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>; then Artin–Schreier–Witt theory associates to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\vec {a}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> a totally ramified cyclic extension L/K of degree <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In the case where k is finite, several authors have used class field theory to explicitly compute the upper ramification breaks of L/K in terms of the valuations of the components of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\vec {a}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation>. In this note, we use a direct method to show that these formulas remain valid when k is an arbitrary perfect field.

Artin–Schreier–Witt extensions and ramification breaks

期刊论文
发表日期：

2026年4月24日

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G. Griffith Elder,
Kevin Keating

摘要

Let $K=k((t))$ $K = k ((t))$ be a local field of characteristic $p>0$ $p > 0$ with perfect residue field k. Let $\vec {a}=(a_0,a_1,\dots ,a_{n-1})\in W_n(K)$ $\vec{a} = (a_{0}, a_{1}, \dots, a_{n - 1}) \in W_{n} (K)$ be a Witt vector of length n. Assume that $\vec {a}$ $\vec{a}$ is “reduced”, and that $v_K(a_0)<0$ $v_{K} (a_{0}) < 0$ ; then Artin–Schreier–Witt theory associates to $\vec {a}$ $\vec{a}$ a totally ramified cyclic extension L/K of degree $p^n$ $p^{n}$ . In the case where k is finite, several authors have used class field theory to explicitly compute the upper ramification breaks of L/K in terms of the valuations of the components of $\vec {a}$ $\vec{a}$ . In this note, we use a direct method to show that these formulas remain valid when k is an arbitrary perfect field.