Artin-Schreier Theory

The Artin-Schreier theory belongs in the mathematics to field theory . For bodies with positive characteristics , it describes Galois Abelian expansions of the exponent and thus complements the sorrow theory . It is named after Emil Artin and Otto Schreier . ${\displaystyle p}$ ${\displaystyle p}$

Motivation: cyclical extensions of degree p

Be a body of characteristic . The starting point of the Artin-Schreier theory is the Artin-Schreier polynomial ${\displaystyle K}$ ${\displaystyle p}$

${\displaystyle f_{a}(X)=X^{p}-X-a}$

for a . From Fermat's little theorem, or more abstractly from the properties of the Frobenius homomorphism, it follows: For is . This results in: If a zero of is in an expansion field of , then the other zeros are . Has no root in , therefore it is irreducible and the extension field is Galois with Galois group generated by . ${\displaystyle a\in K}$ ${\displaystyle c\in \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle f_{a}(X+c)=f_{a}(X)}$ ${\displaystyle \omega }$ ${\displaystyle f_{a}(X)}$ ${\displaystyle K}$ ${\displaystyle \omega +1,\omega +2,\dots ,\omega +(p-1)}$ ${\displaystyle f_{a}(X)}$ ${\displaystyle K}$ ${\displaystyle K(\omega )/K}$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle \omega \mapsto \omega +1}$

Conversely, let be a Galois extension of degree and a producer of the Galois group. After the normal base rate one exists , so that a basis of as - vector space is. After construction is the track ${\displaystyle L/K}$ ${\displaystyle p}$ ${\displaystyle \sigma }$ ${\displaystyle x\in L}$ ${\displaystyle x,\sigma x,\dots ,\sigma ^{p-1}x}$ ${\displaystyle L}$ ${\displaystyle K}$

${\displaystyle {\text{Spur}}_{L/K}(x)=x+\sigma x+\dots +\sigma ^{p-1}x}$

not 0. Put

${\displaystyle \omega =-{\frac {1}{{\text{Spur}}_{L/K}(x)}}\sum _{k=1}^{p-1}k\cdot \sigma ^{k}x}$

Then

${\displaystyle {\begin{aligned}\sigma (\omega )&=-{\frac {1}{{\text{Spur}}_{L/K}(x)}}\sum _{k=1}^{p-1}k\cdot \sigma ^{k+1}x\\&=-{\frac {1}{{\text{Spur}}_{L/K}(x)}}\left(\sum _{k=0}^{p-1}(k+1)\cdot \sigma ^{k+1}x-\sum _{k=0}^{p-1}\sigma ^{k+1}x\right)\\&=-{\frac {1}{{\text{Spur}}_{L/K}(x)}}\sum _{k=1}^{p}k\cdot \sigma ^{k}x+1\\&=\omega +1\end{aligned}}}$

So is

${\displaystyle \sigma (\omega ^{p}-\omega )=(\sigma \omega )^{p}-\sigma \omega =(\omega +1)^{p}-(\omega +1)=\omega ^{p}-\omega }$

This is invariant under the Galois group, so it is in . ${\displaystyle a=\omega ^{p}-\omega }$ ${\displaystyle K}$

The element so constructed depends on the choice of , but in a controlled way: If another element is with , then is , thus is with an element , and ${\displaystyle a\in K}$ ${\displaystyle x}$ ${\displaystyle \omega _{1}\in L}$ ${\displaystyle \sigma \omega _{1}=\omega _{1}+1}$ ${\displaystyle \sigma (\omega -\omega _{1})=(\omega +1)-(\omega _{1}+1)=\omega -\omega _{1}}$ ${\displaystyle \omega _{1}=\omega +d}$ ${\displaystyle d\in K}$

${\displaystyle \omega _{1}^{p}-\omega _{1}=(\omega +d)^{p}-(\omega +d)=\omega ^{p}+d^{p}-\omega -d=a+(d^{p}-d)}$

Consequently the remainder class of modulo is uniquely determined. ${\displaystyle a}$ ${\displaystyle \{d^{p}-d:d\in K\}}$

Results

Be a body of characteristic . ${\displaystyle K}$ ${\displaystyle p>0}$

• Be . The mapping which assigns the decay field of the polynomial to an element induces a bijection of onto the set of isomorphism classes of Galois extensions of from degree . ${\displaystyle \wp K=\{d^{p}-d:d\in K\}}$ ${\displaystyle a\in K}$ ${\displaystyle X^{p}-X-a}$ ${\displaystyle (K/\wp K)\setminus \{0\}}$ ${\displaystyle K}$ ${\displaystyle p}$

The more general version by Ernst Witt reads:

• Let be a separable closure of and the additive group homomorphism . Then there is the following explicit bijection between the set of subgroups of and the set of (not necessarily finite) Abelian expansions of from exponent (i.e. for every element of the Galois group ): Let a subgroup of be identified with its archetype in . Then is the corresponding Abelian expansion of the exponent . For finite subgroups is . The reverse mapping assigns the group to an extension . ${\displaystyle K^{\text{sep}}}$ ${\displaystyle K}$ ${\displaystyle \wp :K^{\text{sep}}\to K^{\text{sep}}}$ ${\displaystyle x\mapsto x^{p}-x}$ ${\displaystyle K/\wp K}$ ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma ^{p}={\text{id}}}$ ${\displaystyle \Delta \subseteq K/\wp K}$ ${\displaystyle K}$ ${\displaystyle K(\wp ^{-1}(\Delta ))/K}$ ${\displaystyle p}$ ${\displaystyle \Delta \subseteq K/\wp K}$ ${\displaystyle [K(\wp ^{-1}(\Delta )):K]=|\Delta |}$ ${\displaystyle L/K}$ ${\displaystyle (K\cap \wp L)/\wp K}$

Galois cohomological interpretation

Continue to be a body of characteristic , a separable closure from and . Also be the absolute Galois group of . The polynomial is separable for each because its derivative is. Therefore homomorphism is surjective. Its core is . So you get a short exact sequence of - modules : ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle K^{\text{sep}}}$ ${\displaystyle K}$ ${\displaystyle \wp \colon K^{\text{sep}}\to K^{\text{sep}},\ x\mapsto x^{p}-x}$ ${\displaystyle G_{K}={\text{Gal}}(K^{\text{sep}}/K)}$ ${\displaystyle K}$ ${\displaystyle X^{p}-X-a}$ ${\displaystyle a\in K^{\text{sep}}}$ ${\displaystyle pX^{p-1}-1=-1}$ ${\displaystyle \wp \colon K^{\text{sep}}\to K^{\text{sep}}}$ ${\displaystyle \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle G_{K}}$

${\displaystyle 0\to \mathbb {Z} /p\mathbb {Z} \to K^{\text{sep}}{\stackrel {\wp }{\longrightarrow }}K^{\text{sep}}\to 0}$

It induces a long exact sequence in Galois cohomology

${\displaystyle 0\to \mathbb {Z} /p\mathbb {Z} \to K{\stackrel {\wp }{\longrightarrow }}K\to {\text{Hom}}(G_{K},\mathbb {Z} /p\mathbb {Z} )\to H^{1}(G_{K},K^{\text{sep}})=0}$

The following was used:

• ${\displaystyle H^{0}(G_{K},K^{\text{sep}})=K}$
• ${\displaystyle H^{1}(G_{K},\mathbb {Z} /p\mathbb {Z} )={\text{Hom}}(G_{K},\mathbb {Z} /p\mathbb {Z} )}$(continuous homomorphisms), because trivial operates on ${\displaystyle G_{K}}$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$
• ${\displaystyle H^{1}(G_{K},K^{\text{sep}})=0}$, because is over all finite Galois extensions of . With a generalization of the above argument with the normal basis theorem one can show. ${\displaystyle H^{1}(G_{K},K^{\text{sep}})=\varinjlim H^{1}({\text{Gal}}(L/K),L)}$ ${\displaystyle K}$ ${\displaystyle H^{1}({\text{Gal}}(L/K),L)=0}$

For the consideration of extensions of degree the general statement is not necessary: ​​Let be a Galois extension of degree . Then is , and by concatenation with the projection one obtains a homomorphism . With the embedding you get a 1-cocycle , which is already in the subgroup . The element constructed above has the property for all , so it is a 1-Korand. The general group- cohomological construction shows that there is an archetype of below the connection homomorphism . ${\displaystyle p}$ ${\displaystyle L/K}$ ${\displaystyle p}$ ${\displaystyle {\text{Gal}}(L/K)\cong \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle G_{K}\to {\text{Gal}}(L/K)}$ ${\displaystyle h\colon G_{K}\to \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} \to K^{\text{sep}}}$ ${\displaystyle c\in H^{1}(G_{K},K^{\text{sep}})}$ ${\displaystyle H^{1}({\text{Gal}}(L/K),L)}$ ${\displaystyle \omega \in L}$ ${\displaystyle c(\sigma )=\sigma \omega -\omega }$ ${\displaystyle \sigma \in {\text{Gal}}(L/K)}$ ${\displaystyle c}$ ${\displaystyle \wp (\omega )}$ ${\displaystyle h}$

If the reverse is given, one can choose an archetype and the homomorphism is . The core of and correspond to each other under the Galois correspondence . ${\displaystyle a\in K}$ ${\displaystyle \omega \in \wp ^{-1}(a)}$ ${\displaystyle h\colon G_{K}\to \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle h(\sigma )=\sigma \omega -\omega }$ ${\displaystyle h}$ ${\displaystyle L=K(\omega )}$

So the isomorphism resulting from the long exact sequence is identical to the explicit construction explained above. ${\displaystyle K/\wp K\to {\text{Hom}}(G_{K},\mathbb {Z} /p\mathbb {Z} )}$

For the more general statement about subgroups, one still has to identify subgroups of with extensions of the exponent : A subgroup corresponds to the fixed field of , an Abelian extension of the exponent corresponds to the subgroup of homomorphisms that factor over the quotient . ${\displaystyle {\text{Hom}}(G_{K},\mathbb {Z} /p\mathbb {Z} )}$ ${\displaystyle p}$ ${\displaystyle \Delta \subseteq {\text{Hom}}(G_{K},\mathbb {Z} /p\mathbb {Z} )}$ ${\displaystyle \textstyle \bigcap _{h\in \Delta }\ker(h)}$ ${\displaystyle L/K}$ ${\displaystyle p}$ ${\displaystyle G_{K}\to {\text{Gal}}(L/K)}$

Artin-Schreier-Symbol and class field theory

The Artin-Schreier symbol is a supplement to the power remainder symbol and, like this, serves to explicitly describe the local reciprocity mapping and thus leads to a partial statement of the existence theorem of the local class field theory . Let be a local field of the characteristic , i.e. H. isomorphic to a formal Laurent series body for a potency . The Artin-Schreier symbol arises from the cohomological pairing ${\displaystyle K}$ ${\displaystyle p>0}$ ${\displaystyle \mathbb {F} _{q}((T))}$ ${\displaystyle q=p^{e}}$

${\displaystyle K/\wp K\times G_{K}\to \mathbb {Z} /p\mathbb {Z} }$

by concatenation with the reciprocity mapping . Is and with and , then: ${\displaystyle ({-},{*}/K)\colon K^{*}\to G_{K}^{\text{ab}}}$ ${\displaystyle a\in K}$ ${\displaystyle \omega \in K^{\text{sep}}}$ ${\displaystyle \wp (\omega )=a}$ ${\displaystyle b\in K^{*}}$

${\displaystyle [a,b)=(b,{*}/K)\omega -\omega }$

The Artin-Schreier symbol induces a non- degenerate bilinear form

${\displaystyle K/\wp K\times K^{*}/(K^{*})^{p}\to \mathbb {Z} /p\mathbb {Z} }$

Other features:

• It applies precisely when a norm is expanding . ${\displaystyle [a,b)=0}$ ${\displaystyle b}$ ${\displaystyle K(\omega )/K}$
• It applies to everyone . ${\displaystyle [a,a)=0}$ ${\displaystyle a\in K^{*}}$

The Artin-Schreier symbol has the following explicit description: Let be a symbol, the one-dimensional vector space spanned by as well ${\displaystyle dT}$ ${\displaystyle \Omega =\mathbb {F} _{q}((T))\cdot dT}$ ${\displaystyle dT}$

${\displaystyle d\colon \mathbb {F} _{q}((T))\to \Omega ,\ \ \sum a_{n}T^{n}\mapsto \left(\sum a_{n}\cdot nT^{n-1}\right)dT}$

and the residual map

${\displaystyle {\text{res}}\colon \Omega \to \mathbb {F} _{q},\ \ \left(\sum a_{n}T^{n}\right)dT\mapsto a_{-1}}$

(The construction is independent of the isomorphism .) For and is then: ${\displaystyle K\cong \mathbb {F} _{q}((T))}$ ${\displaystyle a\in K}$ ${\displaystyle b\in K^{*}}$

${\displaystyle [a,b)={\text{Spur}}_{\mathbb {F} _{q}/\mathbb {F} _{p}}\ {\text{res}}\left(a\cdot {\frac {db}{b}}\right)}$

From this formula it can be shown that the Artin-Schreier symbol has not degenerated as claimed. It follows that an element in which is in the norm group for every Galois expansion of degree is a -th power. It follows from this that the intersection of all groups of norms is trivial, an essential step (depending on the approach) in the proof of the local existence theorem. ${\displaystyle K^{*}}$ ${\displaystyle L/K}$ ${\displaystyle p}$ ${\displaystyle N_{L/K}L^{*}}$ ${\displaystyle p}$

The local Artin-Schreier symbols can also be used as a global pairing

${\displaystyle \mathbb {A} _{K}/\wp \mathbb {A} _{K}\times I_{K}/I_{K}^{p}\to \mathbb {Z} /p\mathbb {Z} }$

(where the needle ring and the Idelgruppe ) together and use to prove the global existence theorem a function field. ${\displaystyle \mathbb {A} _{K}}$ ${\displaystyle I_{K}=\mathbb {A} _{K}^{*}}$

Geometric perspective

At the center of the geometric consideration is the Artin-Schreier morphism

${\displaystyle \wp =F-1:\mathbb {G} _{a}\to \mathbb {G} _{a}}$

which can be understood as the long isogeny for the additive group ( is the relative Frobenius morphism ). is a (connected and therefore not trivial) étale Galois cover with group . The existence of shows that the geometric étale fundamental group of the affine line is not trivial, in contrast to the situation in characteristic 0. ${\displaystyle \mathbb {G} _{a}=\mathbb {A} ^{1}}$ ${\displaystyle F}$ ${\displaystyle \wp }$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle \wp }$

A body element corresponds to a morphism , and the fiber from over is either the trivial torsor or the Artin-Schreier expansion of, as defined by the polynomial . ${\displaystyle a\in K}$ ${\displaystyle a:{\text{Spec }}K\to \mathbb {G} _{a}}$ ${\displaystyle \wp }$ ${\displaystyle a}$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle X^{p}-X-a}$ ${\displaystyle K}$

Sheaves associated with the Artin-Schreier torsor are relevant for the Fourier-Deligne transformation .

Artin-Schreier-Witt theory

The theory outlined here generalizes the Artin-Schreier theory to extensions whose exponent is a power of . It is the content of Witt's work in which he introduces the Witt vectors. The first part is a general statement about Abelian extensions of fields of the characteristic , the second part an explicit description of part of the local class field theory in the case of function fields . ${\displaystyle p}$ ${\displaystyle p}$

Let again be a body of characteristic , a separable closure of and the absolute Galois group of . Let the group of -typical Witt vectors be the length and the Frobenius homomorphism ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle K^{\text{sep}}}$ ${\displaystyle K}$ ${\displaystyle G_{K}={\text{Gal}}(K^{\text{sep}}/K)}$ ${\displaystyle K}$ ${\displaystyle W_{n}}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle F}$

${\displaystyle (x_{0},\dots ,x_{n-1})\mapsto (x_{0}^{p},\dots ,x_{n-1}^{p})}$

With

${\displaystyle \wp :W_{n}(K^{\text{sep}})\to W_{n}(K^{\text{sep}}),\ \ x\mapsto F(x)-x}$

is

${\displaystyle 0\to \mathbb {Z} /p^{n}\mathbb {Z} \to W_{n}(K^{\text{sep}}){\stackrel {\wp }{\longrightarrow }}W_{n}(K^{\text{sep}})\to 0}$

an exact sequence of modules where was used. The Galois cohomology disappears because the quotients with respect to the -filtering are isomorphic and holds (see above). So is , and as above one obtains a correspondence between Abelian extensions, whose exponent is a divisor of , and subgroups of . ${\displaystyle G_{K}}$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} \cong W_{n}(\mathbb {F} _{p})}$ ${\displaystyle H^{1}(G_{K},W_{n})}$ ${\displaystyle V}$ ${\displaystyle K^{\text{sep}}}$ ${\displaystyle H^{1}(G_{K},K^{\text{sep}})=0}$ ${\displaystyle H^{1}(G_{K},\mathbb {Z} /p^{n}\mathbb {Z} )\cong W_{n}(K)/\wp W_{n}(K)}$ ${\displaystyle p^{n}}$ ${\displaystyle W_{n}(K)/\wp W_{n}(K)}$

Be a local body ( formal Laurent series ). For a Witt vector and a body element , Witt defines a central simple algebra that consists of and the commuting elements with the relations ${\displaystyle K\cong \mathbb {F} _{q}((T))}$ ${\displaystyle a\in W_{n}(K)}$ ${\displaystyle b\in K^{*}}$ ${\displaystyle A_{[a,b)}}$ ${\displaystyle u}$ ${\displaystyle v_{0},\dots ,v_{n-1}}$

${\displaystyle u^{p^{n}}=b,\ \wp (v)=a,\ v^{u}=v+1}$

is produced. It is calculated with as a Witt vector, and stands for the Witt vector . Be with and , also the reciprocity mapping. The Artin-Schreier-Witt symbol is defined as ${\displaystyle v=(v_{0},\dots ,v_{n-1})}$ ${\displaystyle v^{u}}$ ${\displaystyle (uv_{0}u^{-1},\dots ,uv_{n-1}u^{-1})}$ ${\displaystyle \omega \in W_{n}(K^{\text{sep}})}$ ${\displaystyle \wp (\omega )=a}$ ${\displaystyle L=K(\omega )=K(\omega _{0},\dots ,\omega _{n-1})}$ ${\displaystyle ({-},L/K)}$

${\displaystyle [a,b)=(b,L/K)(\omega )-\omega \in W_{n}(\mathbb {F} _{p})\cong {\tfrac {1}{p^{n}}}\mathbb {Z} /\mathbb {Z} \subset \mathbb {Q} /\mathbb {Z} }$

The Artin-Schreier-Witt symbol is a non-degenerate bilinear pairing

${\displaystyle W_{n}(K)/\wp W_{n}(K)\times K^{*}/(K^{*})^{p^{n}}\to \mathbb {Q} /\mathbb {Z} }$

It is exactly when . The value of the symbol is equal to the invariant of central simple algebra . Witt also gives a description of the invariant as a residue continued on Witt vectors of Laurent series . ${\displaystyle [a,b)=0}$ ${\displaystyle b\in N_{L/K}(K^{*})}$ ${\displaystyle [a,b)={\text{inv}}(A_{[a,b)})}$

literature

• Jürgen Neukirch, Alexander Schmidt, Kay Wingberg: Cohomology of Number Fields . Springer, Berlin 2000, ISBN 3-540-66671-0 , chap. VI §1.
• Peter Roquette: Class Field Theory in Characteristic p, its Origin and Development . In: Class Field Theory, its Centenary and Prospect . Math. Soc. Japan, Tokyo 2001, p. 549-631 .
• J.-P. Serre : Local Fields . Springer, Berlin 1979, ISBN 3-540-90424-7 .

Footnotes

1. ↑ The original work is: Emil Artin, Otto Schreier: A designation of real closed bodies . In: Treatises from the mathematical seminar at the University of Hamburg . tape 5 , no. 1 , 1927, pp. 225-231 , doi : 10.1007 / BF02952522 .
2. ↑ Roquette 2001, chap. 7.2. The original work is: Ernst Witt: The existence theorem for Abelian function bodies . In: Journal for pure and applied mathematics . tape 173 , 1935, pp. 34-51 .
3. ^ Formula given for the first time by Hermann Ludwig Schmid , see Roquette 2001, chap. 7.1. The original work is: Hermann Ludwig Schmid: About the reciprocity law in relatively cyclic algebraic function fields with finite constant fields . In: Mathematical Journal . tape 40 , 1935, pp. 91-109 .
4. ↑ Serre 1979, XIV §6
5. ^ André Weil: Basic Number Theory . 3. Edition. Springer, New York 1974, ISBN 0-387-06935-6 , chap. XIII §7. Shokichi Iyanaga: The Theory of Numbers . North-Holland, Amsterdam 1975, ISBN 0-444-10678-2 , chap. V §4.
6. ^ Reinhardt Kiehl, Rainer Weissauer: Weil Conjectures, Perverse Sheaves and I-adic Fourier Transform . Springer, Berlin 2001, ISBN 3-540-41457-6 .
7. ↑ Ernst Witt: Cyclic fields and algebras of the characteristic p of degree p ⁿ . Structure of discretely valued perfect bodies with a perfect residue class field of characteristic p . In: J. Reine Angew. Math. Band 176 , 1936, pp. 126-140 .
8. ^ Nathan Jacobson: Basic Algebra II . WH Freeman and Company, San Francisco 1980, ISBN 0-7167-1079-X , chap. 8.11. Nicolas Bourbaki: Eléments de mathématique. Algèbre commutative. Chapitres 8 and 9 . Springer, Berlin 2006, ISBN 978-3-540-33942-7 , chap. IX §1 Ex. 19-21.
9. ↑ See also: Lara Thomas: Ramification groups in Artin-Schreier-Witt extensions . In: Journal de Théorie des Nombres de Bordeaux . tape 17 , no. 2 , 2005, p. 689-720 ( online ).