mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Artin–Schreier theory is a branch of

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

, specifically a positive characteristic analogue of

Kummer theory In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer aro ...

, for Galois

extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ex ...

of degree equal to the characteristic ''p''. introduced Artin–Schreier theory for extensions of prime degree ''p'', and generalized it to extensions of prime power degree ''p''^''n''. If ''K'' is a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of characteristic ''p'', a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, any

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

of the form :

X^p - X - \alpha,\,

for

\alpha

in ''K'', is called an ''Artin–Schreier polynomial''. When

\alpha\neq \beta^p-\beta

for all

\beta \in K

, this polynomial is

irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

in ''K'' 'X'' and its

splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...

over ''K'' is a

cyclic extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...

of ''K'' of degree ''p''. This follows since for any root ''β'', the numbers ''β'' + ''i'', for

1\le i\le p

, form all the roots—by

Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...

—so the splitting field is

K(\beta)

. Conversely, any Galois extension of ''K'' of degree ''p'' equal to the characteristic of ''K'' is the splitting field of an Artin–Schreier polynomial. This can be proved using additive counterparts of the methods involved in

, such as

Hilbert's theorem 90 In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' is an extension of fi ...

and additive

Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...

. These extensions are called ''Artin–Schreier extensions''. Artin–Schreier extensions play a role in the theory of

solvability by radicals In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

, in characteristic ''p'', representing one of the possible classes of extensions in a solvable chain. They also play a part in the theory of

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...

and their isogenies. In characteristic ''p'', an isogeny of degree ''p'' of abelian varieties must, for their function fields, give either an Artin–Schreier extension or a

purely inseparable extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...

Artin–Schreier–Witt extensions

There is an analogue of Artin–Schreier theory which describes cyclic extensions in characteristic ''p'' of ''p''-power degree (not just degree ''p'' itself), using

Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of orde ...

s, developed by .

References

* * Section VI.6 * Section VI.1 * {{DEFAULTSORT:Artin-Schreier theory Galois theory