Worry theory

In the mathematical sub-area of body theory , worry theory describes certain body extensions that are obtained through the adjunction of -ter roots of elements of the basic body. The theory was originally developed by Ernst Eduard Kummer while studying the Fermat conjecture in the 1840s. ${\ displaystyle n}$

The main statements of the theory do not depend on the special basic body, only its characteristics must not be a factor of . Worry theory plays a fundamental role in class field theory ; in general, it is important for understanding Abelian extensions; it says that cyclical extensions can be obtained by pulling roots, provided that the basic body contains enough roots of unity. ${\ displaystyle n}$

Sorrow extensions

definition

Be a natural number. An expansion of sorrow is an expansion of the body to which the following applies: ${\ displaystyle n> 1}$ ${\ displaystyle L / K}$

${\ displaystyle K}$ contains various -th roots of unity , i.e. the zeros of the polynomial . ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle X ^ {n} -1}$

{\ displaystyle L / K}

has an Abelian Galois group from the exponent . The latter means that applies to all elements of the Galois group and is minimal with this property.

{\ displaystyle n}

{\ displaystyle \ sigma}

{\ displaystyle \ sigma ^ {n} = \ operatorname {Id}}

{\ displaystyle n}

Examples

If , the first condition is always fulfilled, if it does not have the characteristic 2, the two roots of unity are 1 and . In this case, sorrow expansions are initially square expansions , with a non-square element of being. The formula for solving quadratic equations shows that every degree 2 expansion has this shape. Likewise, sorrow extensions for are biquadratic (by the addition of two square roots) and, more generally, multiquatric (by the addition of several square roots). If the characteristic has 2, there are no extensions of grief, since the equation applies in characteristic 2, i.e. there are no two different roots of unity. ${\ displaystyle n = 2}$ ${\ displaystyle K}$ ${\ displaystyle -1}$ ${\ displaystyle L = K ({\ sqrt {a}})}$ ${\ displaystyle a}$ ${\ displaystyle K}$ ${\ displaystyle n = 2}$ ${\ displaystyle K}$ ${\ displaystyle -1 = 1}$

For there are no sorrow extensions of the rational numbers , since not all three third roots of unity are rational. Let be any rational number that is not a third power and the decay field of over . If and are zeros of this cubic polynomial, then applies . Since the cubic polynomial is also separable , it has three different zeros. This means that the two nontrivial third roots of unity, namely and , lie in , so that it has a lower body that contains the three roots of unity. Then there is an extension of sorrow. ${\ displaystyle n = 3}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle a}$ ${\ displaystyle L}$ ${\ displaystyle X ^ {3} -a}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta}$ ${\ displaystyle (\ alpha / \ beta) ^ {3} = \ alpha ^ {3} / \ beta ^ {3} = a / a = 1}$ ${\ displaystyle \ alpha / \ beta}$ ${\ displaystyle \ beta / \ alpha}$ ${\ displaystyle L}$ ${\ displaystyle L}$ ${\ displaystyle K}$ ${\ displaystyle L / K}$

If, more generally, contains various -th roots of unity, from which it already follows that the characteristic of is not a divisor of , then one obtains an extension of sorrow by the addition of a -th root of an element of to the body . Your degree is a factor of . As the decay field of the polynomial , the Kummer expansion is automatically Galois with a cyclic Galois group of the order . ${\ displaystyle K}$ ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle K}$ ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle a}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle m}$ ${\ displaystyle n}$ ${\ displaystyle X ^ {n} -a}$ ${\ displaystyle m}$

Worry theory

The worry theory makes statements in the opposite direction. If a body contains several -th roots of unity, it says that every cyclic extension of the degree can be obtained by drawing a -th root. If one denotes the multiplicative group of the non-zero elements of the body , then the cyclic expansions of from degree , which are in a fixed algebraic closure, are in bijection with the cyclic subgroups of , i.e. the factor group from after the -th powers. ${\ displaystyle K}$ ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle K}$ ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle K ^ {\ times}}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle n}$ ${\ displaystyle K ^ {\ times} / (K ^ {\ times}) ^ {n}}$ ${\ displaystyle K ^ {\ times}}$ ${\ displaystyle n}$

The bijection can be specified explicitly: A cyclic subgroup is assigned the extension that results from the adjunction of all -th roots of elements . ${\ displaystyle \ Delta \ subseteq K ^ {\ times} / (K ^ {\ times}) ^ {n}}$ ${\ displaystyle K (\ Delta ^ {1 / n})}$ ${\ displaystyle n}$ ${\ displaystyle \ Delta}$ ${\ displaystyle K}$

Conversely, one assigns the subgroup to the expansion of sorrow . ${\ displaystyle L / K}$ ${\ displaystyle \ Delta = K ^ {\ times} \ cap (L ^ {\ times}) ^ {n}}$

If this bijection assigns the group and the body extension to one another, there is an isomorphism which is given by . Here stands for the group of -th roots of unity and for any -th root of . ${\ displaystyle \ Delta}$ ${\ displaystyle L / K}$ ${\ displaystyle \ Delta \ cong \ operatorname {Hom} (\ operatorname {Gal} (L / K), \ mu _ {n})}$ ${\ displaystyle a \ mapsto (\ sigma \ mapsto {\ tfrac {\ sigma (\ alpha)} {\ alpha}})}$ ${\ displaystyle \ mu _ {n}}$ ${\ displaystyle n}$ ${\ displaystyle \ alpha}$ ${\ displaystyle n}$ ${\ displaystyle a \ in \ Delta}$

Generalizations

The above correspondence continues to a bijection between subgroups and Abelian extensions of the exponent . This general version was first given by Ernst Witt . ${\ displaystyle \ Delta \ subseteq K ^ {\ times} / (K ^ {\ times}) ^ {n}}$ ${\ displaystyle n}$

In characteristics there is an analogous theory for cyclic extensions of the degree , the Artin-Schreier theory . A generalization for Abelian expansions of the exponent also comes from Witt. It uses the Witt vectors introduced in the same work . ${\ displaystyle p> 0}$ ${\ displaystyle p}$ ${\ displaystyle p ^ {n}}$

Footnotes

^ 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 . The original work by Witt is: Ernst Witt: The Existence Theorem for Abelian function bodies . In: Journal for pure and applied mathematics . tape 173 , 1935, pp. 34-51 .
↑ 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: Journal for pure and applied mathematics . tape 176 , 1936, pp. 126-140 .

swell

Jürgen Neukirch : class field theory . Bibliographical Institute, Mannheim 1986.
LV Kuz'min: Kummer extension . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

[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 . The original work by Witt is: Ernst Witt: The Existence Theorem for Abelian function bodies . In: Journal for pure and applied mathematics . tape 173 , 1935, pp. 34-51 .

[2] 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: Journal for pure and applied mathematics . tape 176 , 1936, pp. 126-140 .