Square class

In algebra , square classes are the equivalence classes of a certain equivalence relation , the quadratic equivalence in a commutative group . They are then the subsidiary classes of the subgroup of the squares in that group. The concept of square classes and square equivalence is applied, among other things

in linear algebra for the affine classification of quadrics in an affine space over any field ,
in projective geometry in the projective classification of projective quadrics in a projective space over any body,
in synthetic geometry

when examining orthogonality relations, see Pre-Euclidean level ,
when investigating possible arrangements of bodies (→ see Pythagorean bodies and, as a special case, Euclidean bodies ),

in number theory when studying quadratic Diophantine equations .

Square classes are also defined more generally in the literature, whereby the consequences of the common, group-theoretical term mostly emerge as the essential core of the more general concept.

Definitions

The general definition of a "quadratic relation" has the advantage that it can always be used meaningfully if this definition leads to an equivalence relation. The group-theoretical definition shows that the quadratic relation is an equivalence relation at least for commutative groups, and the quadratic classes are therefore actually a division of the group into secondary classes of a subgroup. In this special case all theorems and properties for secondary classes of the normal divisors of any and the subgroups of an Abelian group can be applied to square classes .

general definition

Let be a set with the two-place linkage and a non-empty subset that is closed with respect to this linkage. Then a two-digit relation is introduced through the definition ${\ displaystyle M}$ ${\ displaystyle \ cdot}$ ${\ displaystyle A}$ ${\ displaystyle M}$ ${\ displaystyle \ sim}$

${\ displaystyle m_ {1} \ sim m_ {2}}$ if there are elements such that is. ${\ displaystyle a_ {1}, a_ {2} \ in A}$ ${\ displaystyle m_ {1} \ cdot a_ {1} ^ {2} = m_ {2} \ cdot a_ {2} ^ {2}}$

Now applies:

By definition, the relation is always reflexive and symmetrical .
It is then safe transitive if the link associatively on and on commutative is. ${\ displaystyle M}$ ${\ displaystyle A}$

The following, weaker conditions are sufficient for transitivity: For elements always exist such that ${\ displaystyle m \ in M; \; a, b \ in A}$ ${\ displaystyle c, d \ in A}$

${\ displaystyle (m \ cdot a ^ {2}) \ cdot b ^ {2} = m \ cdot c ^ {2}}$ (Weakening of associativity) and
${\ displaystyle a ^ {2} \ cdot b ^ {2} = d ^ {2} \;}$ (Weakening of the commutativity) applies.

In all cases in which the relation is transitive, i.e. an equivalence relation, two elements of which satisfy the relation are called quadratically equivalent (in the broader sense) with respect to the subset . Each equivalence class of this relation that is an element of containing means square class (in the narrow sense) of respect . ${\ displaystyle M}$ ${\ displaystyle A}$ ${\ displaystyle A}$ ${\ displaystyle M}$ ${\ displaystyle A}$

Group theoretical definition

Be a commutative group. Then is the square figure ${\ displaystyle (G, \ cdot)}$

{\ displaystyle q \ colon G \ rightarrow G; g \ mapsto g ^ {2}}

a group homomorphism . Its image, ie the set of “squares”, is a subgroup of and the secondary classes of this subgroup are called square classes of . ${\ displaystyle G ^ {2} = \ lbrace g ^ {2} | g \ in G \ rbrace}$ ${\ displaystyle G}$ ${\ displaystyle G}$

This is the special case of the general definition when there is a set. ${\ displaystyle M = A = G}$

If the square map is surjective , there is only one class of squares, which then includes the whole group. This case occurs for finite groups if and only if the mapping is injective and therefore, according to Lagrange's theorem and Sylow's theorems , if the order of the group is odd and therefore no element has an even order .

More generally, the number of square classes is the index of the squares in . ${\ displaystyle (G: G ^ {2})}$ ${\ displaystyle G}$

Square classes in commutative rings

body

In a body , the quadratic equivalence with respect to the multiplicative group is usually referred to as the quadratic equivalence. The equivalence class (in the broader sense) of 0 consists only of the zero element, all others are square classes of in the sense of the general definition and of in the narrower sense and in the sense of the group-theoretical definition. ${\ displaystyle (K, +, \ cdot)}$ ${\ displaystyle (K ^ {*}, \ cdot)}$ ${\ displaystyle K}$ ${\ displaystyle (K ^ {*}, \ cdot)}$

Health scope

In an integrity domain (with one element), as in a body, quadratic equivalence with respect to the commutative monoid that can be shortened is generally referred to as the quadratic equivalence. Here, too, all equivalence classes are except subsets of and thus square classes of (in the narrower sense). ${\ displaystyle (R, +, \ cdot)}$ ${\ displaystyle (R \ setminus \ lbrace 0 \ rbrace, \ cdot)}$ ${\ displaystyle \ lbrace 0 \ rbrace}$ ${\ displaystyle A = R \ setminus \ lbrace 0 \ rbrace}$ ${\ displaystyle R}$

In addition, the quadratic equivalence here is compatible with the embedding of the integrity area in its quotient field : Two elements of the integrity area are squarely equivalent in the ring if they (more precisely: the images of these elements under the embedding) also in the quotient field (there also in the sense of group-theoretical definition) are quadratically equivalent. In addition, each square class of the quotient field contains “whole” elements, that is, embedded images of elements of the integrity area . ${\ displaystyle \ operatorname {Quot} (R)}$ ${\ displaystyle R}$

Examples

The field of real numbers contains exactly two square classes, namely the set of positive and negative real numbers. This is more general for any Euclidean body .

The field of complex numbers contains only one class of squares, namely . The same applies to every algebraically closed field. ${\ displaystyle \ mathbb {C} ^ {*} = \ mathbb {C} \ setminus \ lbrace 0 \ rbrace}$

The integer integrity domain contains infinitely many classes of squares. Two whole numbers (except 0) are squarely equivalent if and only if their product is a square number , i.e. squarely equivalent to 1. The square-free numbers form a system of representatives . ${\ displaystyle \ mathbb {Z}}$

The remainder class field contains only one square class if is and exactly two square classes if is an odd prime number . For the geometry, the following distinction is also important: If the odd prime number is of the form , then −1 and 1 are squared equivalent, for they are in different square classes. (→ See quadratic remainder , quadratic reciprocity law and - for a geometric application - pre-Euclidean plane ). ${\ displaystyle K = \ mathbb {Z} / p \ mathbb {Z}}$ ${\ displaystyle p = 2}$ ${\ displaystyle p}$ ${\ displaystyle p}$ ${\ displaystyle p = 4 \ cdot k + 1, \; k \ in \ mathbb {N}}$ ${\ displaystyle p = 4 \ cdot k + 3, \; k \ in \ mathbb {N} _ {0}}$

All finite fields with characteristic 2 have exactly one square class. Therefore every purely square equation is solvable in these fields and has exactly one double-counting solution due to the Frobenius homomorphism . ${\ displaystyle \ mathbb {F} _ {2 ^ {n}}}$ ${\ displaystyle X ^ {2} + c = 0}$

A non-commutative example results for the quaternion group . Although this group is not commutative, the 4 subclasses of the center are square classes of the group (with respect to the group itself) in the sense of the general definition. Since this group is also a multiplicative group of a quasi-body (→ the quasi-body described in Ternary bodies # Examples of order 9 ), these square classes are of interest in synthetic geometry. For the quasi-body is also the core . ${\ displaystyle Q_ {8} = \ lbrace \ pm 1, \ pm i, \ pm j, \ pm k \ rbrace}$ ${\ displaystyle Z (Q_ {8}) = (Q_ {8}) ^ {2} = \ lbrace \ pm 1 \ rbrace}$ ${\ displaystyle A = Q_ {8}}$ ${\ displaystyle J_ {9}}$ ${\ displaystyle J_ {9}}$ ${\ displaystyle \ lbrace 0 \ rbrace \ cup Z (Q_ {8})}$

literature

Martin Aigner , Dieter Jungnickel (Eds.): Geometries and groups. Proceedings of a colloquium, held at the Freie Universität Berlin, May 1981 . Springer, Berlin / Heidelberg / New York 1981, ISBN 3-540-11166-2 .
Oleg Tomovich Ižboldin, Jean-Pierre Tignol (Ed.): Geometric methods in the algebraic theory of quadratic forms . summer school, Lens, 2000. Springer, Berlin / Heidelberg / New York / Hong Kong / London / Milan / Paris / Tokyo 2000, ISBN 3-540-20728-7 (Lecture notes in mathematics, Vol. 1835).
Helmut Hasse : About the representability of numbers using square forms in the body of rational numbers . In: Journal for pure and applied mathematics . 1923 ( full text at the Göttingen digitization center ).
Hanfried Lenz : Square forms and collineation groups . In: Archives of Mathematics . tape 18 . Hanover 1962, p. 110-119 , doi : 10.1007 / BF01650054 .
Armin Leutbecher: Number Theory: An Introduction to Algebra . Springer, Berlin / Heidelberg / New York 1996, ISBN 3-540-58791-8 .