The Ramification Groups and Different of a Compositum of Artin–schreier Extensions

Wu, Qingquan; Scheidler, Renate

doi:10.1142/s1793042110003617

Cited by 5 publications

(10 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…At a ramified place

p_{2}

of L 2 in L above a place

scriptP

of K such that

v_{scriptP} false(z false) < 0

, one has by the strict triangle inequality that when

α \neq α^{p}

\begin{matrix} true \\ right v_{{frakturp}_{2}} ({\overset{y}{true}}_{1}) & left = trueprefixmin ({} v_{p_{2}} false(a_{1} false), v_{p_{2}} ({false(y_{2} false)}^{k} {false(m_{2} z false)}^{n - k}), v_{p_{2}} true(y_{2}^{n} true)) \\ left = trueprefixmin ({} v_{p_{2}} false(a_{1} false), v_{scriptP} false(z false) false(k + p (n - k) false) false(k \in {1, \dots, n} false), n v_{scriptP} false(z false)) \\ left = v_{P} (z) (1 + p false(n - 1 false)) < 0 \end{matrix}

and coprime to p . Here, we find the same valuation as in [, Corollary 3.10].…”

Section: Standard Formmentioning

confidence: 57%

“…As defined,

{\tilde{y}}_{1}

is an Artin–Schreier generator of

L / L_{2}

in global standard form. We have

\begin{matrix} true \\ right {\tilde{y}}_{1}^{p} - {\tilde{y}}_{1} & left = y_{1}^{p} - y_{1} - m_{1} m_{2}^{- n} {false(y_{2} + m_{2} z false)}^{n} + α y_{2}^{n} \\ left = y_{1}^{p} - y_{1} - m_{1} m_{2}^{- n} \sum_{k = 0}^{n} (\frac{0pt}{n}) {(y_{2})}^{k} {(m_{2} z)}^{n - k} + α y_{2}^{n} \\ left = a_{1} - m_{1} m_{2}^{- n} \sum_{k = 1}^{n} (\frac{0pt}{n}) {(y_{2})}^{k} {(m_{2} z)}^{n - k} + α y_{2}^{n} \\ left = a_{1} - m_{1} m_{2}^{- n} \sum_{k = 1}^{n - 1} (\frac{0pt}{n}) {(y_{2})}^{k} {(m_{2} z)}^{n - k} + false(α - m_{1} m_{2}^{- n} false) y_{2}^{n} . \end{matrix}

By the work of Wu and Scheidler , we know that

L / L<...>

…”

Section: Standard Formmentioning

confidence: 99%

“…By [, Lemma 1.2], the elements

y_{μ}

(μ \in {double-struckF}_{p^{n}})

defined by

y_{μ}^{p} - y_{μ} = μ z

are precisely the Artin–Schreier generators of all of the subextensions of L of degree p over K . For any place

scriptP

of K ramified in L and

μ \neq μ^{'}

, we have for the respective places

P_{μ}

and

P_{μ^{'}}

K (y_{μ})

and

K (y_{μ^{'}})

above

scriptP

that

v_{P_{μ}} false(y_{μ} false) = v_{P_{μ^{'}}} false(y_{μ^{'}} false) = v_{scriptP} false(z false) .

If z is in global standard form, then any places

scriptP

of K are either unramified or fully ramified [, Theorem 3.11]. In particular, it is impossible to associate a partially ramified place with standard form, even locally .…”

Section: Standard Formmentioning

confidence: 99%

See 2 more Smart Citations

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

Marques

Ward

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…At a ramified place

p_{2}

of L 2 in L above a place

scriptP

of K such that

v_{scriptP} false(z false) < 0

, one has by the strict triangle inequality that when

α \neq α^{p}

\begin{matrix} true \\ right v_{{frakturp}_{2}} ({\overset{y}{true}}_{1}) & left = trueprefixmin ({} v_{p_{2}} false(a_{1} false), v_{p_{2}} ({false(y_{2} false)}^{k} {false(m_{2} z false)}^{n - k}), v_{p_{2}} true(y_{2}^{n} true)) \\ left = trueprefixmin ({} v_{p_{2}} false(a_{1} false), v_{scriptP} false(z false) false(k + p (n - k) false) false(k \in {1, \dots, n} false), n v_{scriptP} false(z false)) \\ left = v_{P} (z) (1 + p false(n - 1 false)) < 0 \end{matrix}

and coprime to p . Here, we find the same valuation as in [, Corollary 3.10].…”

Section: Standard Formmentioning

confidence: 57%

“…As defined,

{\tilde{y}}_{1}

is an Artin–Schreier generator of

L / L_{2}

in global standard form. We have

\begin{matrix} true \\ right {\tilde{y}}_{1}^{p} - {\tilde{y}}_{1} & left = y_{1}^{p} - y_{1} - m_{1} m_{2}^{- n} {false(y_{2} + m_{2} z false)}^{n} + α y_{2}^{n} \\ left = y_{1}^{p} - y_{1} - m_{1} m_{2}^{- n} \sum_{k = 0}^{n} (\frac{0pt}{n}) {(y_{2})}^{k} {(m_{2} z)}^{n - k} + α y_{2}^{n} \\ left = a_{1} - m_{1} m_{2}^{- n} \sum_{k = 1}^{n} (\frac{0pt}{n}) {(y_{2})}^{k} {(m_{2} z)}^{n - k} + α y_{2}^{n} \\ left = a_{1} - m_{1} m_{2}^{- n} \sum_{k = 1}^{n - 1} (\frac{0pt}{n}) {(y_{2})}^{k} {(m_{2} z)}^{n - k} + false(α - m_{1} m_{2}^{- n} false) y_{2}^{n} . \end{matrix}

By the work of Wu and Scheidler , we know that

L / L<...>

…”

Section: Standard Formmentioning

confidence: 99%

“…By [, Lemma 1.2], the elements

y_{μ}

(μ \in {double-struckF}_{p^{n}})

defined by

y_{μ}^{p} - y_{μ} = μ z

are precisely the Artin–Schreier generators of all of the subextensions of L of degree p over K . For any place

scriptP

of K ramified in L and

μ \neq μ^{'}

, we have for the respective places

P_{μ}

and

P_{μ^{'}}

K (y_{μ})

and

K (y_{μ^{'}})

above

scriptP

that

v_{P_{μ}} false(y_{μ} false) = v_{P_{μ^{'}}} false(y_{μ^{'}} false) = v_{scriptP} false(z false) .

If z is in global standard form, then any places

scriptP

of K are either unramified or fully ramified [, Theorem 3.11]. In particular, it is impossible to associate a partially ramified place with standard form, even locally .…”

Section: Standard Formmentioning

confidence: 99%

See 1 more Smart Citation

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

Marques

Ward

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…The elementary abelian extension of Galois group Z p × Z p is investigated in [1] and [22]. It should be mentioned that the idea of utilizing transitivity of differents and Hilbert's Different Formula to investigate the ramification groups is used in [9], where the Hasse-Arf property for elementary abelian extensions of function fields is proved.…”

Section: This Relation Is Close If An Intersection Propertymentioning

confidence: 99%

“…Notice that the strictly increasing property and d i ≡ 0 (mod p) are the only two restrictions on the sequence d i of positive integers. See [1] or [22] for a discussion of the type of the extension L/K. Although an explicit construction is not given there, the extension L/K herein is of the same type as described in those two papers.…”

Section: Chapter V Hasse-arf Propertymentioning

confidence: 99%

The ramification group filtrations of certain function field extensions

Castañeda

2015

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

We investigate the ramification group filtration of a Galois extension of function fields, if the Galois group satisfies a certain intersection property. For finite groups, this property is implied by having only elementary abelian Sylow p-subgroups. Note that such groups could be non-abelian. We show how the problem can be reduced to the totally wild ramified case on a p-extension. Our methodology is based on an intimate relationship between the ramification groups of the field extension and those of all degree p sub-extensions. Not only do we confirm that the Hasse-Arf property holds in this setting, but we also prove that the Hasse-Arf divisibility result is the best possible by explicit calculations of the quotients, which are expressed in terms of the different exponents of all those degree p sub-extensions. v ACKNOWLEDGMENTS

show abstract

Holomorphic differentials of Klein four covers

Bleher

Camacho

2023

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

The Ramification Groups and Different of a Compositum of Artin–schreier Extensions

Cited by 5 publications

References 19 publications

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

The ramification group filtrations of certain function field extensions

Holomorphic differentials of Klein four covers

Contact Info

Product

Resources

About