View source for Skew-field

A [[Ring|ring]] in which the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856802.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856803.png" /> are uniquely solvable. In the case of an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) it is sufficient to require the existence of a unit 1 and the unique solvability of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856805.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856806.png" />. A commutative associative skew-field is called a [[Field|field]]. An example of a non-commutative associative skew-field is the skew-field of quaternions, defined as the set of matrices of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856807.png" /></td> </tr></table>

over the field of complex numbers with the usual operations (see [[Quaternion|Quaternion]]). An example of a non-associative skew-field is the [[Cayley–Dickson algebra|Cayley–Dickson algebra]], consisting of all matrices of the same form as above over the skew-field of quaternions. This skew-field is alternative (see [[Alternative rings and algebras|Alternative rings and algebras]]). Any skew-field is a [[Division algebra|division algebra]] either over the field of rational numbers or over a field of residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856808.png" />. The skew-field of quaternions is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856809.png" />-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568010.png" />-dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [[#References|[1]]], and also [[Topological ring|Topological ring]]). The fields of real or complex numbers and the skew-field of quaternions are the only connected locally compact associative skew-fields (see [[#References|[5]]]). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see [[#References|[6]]], [[#References|[8]]]). An associative skew-field is characterized by the property that any non-zero module over it is free. Any non-associative skew-field is finite-dimensional [[#References|[3]]]. A similar result applies to Mal'tsev skew-fields [[#References|[7]]] (see [[Mal'tsev algebra|Mal'tsev algebra]]) and to Jordan skew-fields [[#References|[4]]] (see [[Jordan algebra|Jordan algebra]]). In contrast to the commutative case, not every associative ring without zero divisors can be imbedded in a skew-field (see [[Imbedding of rings|Imbedding of rings]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.F. Adams,   "On the nonexistence of elements of Hopf invariant one"  ''Bulletin Amer. Math. Soc.'' , '''64''' :  5  (1958)  pp. 279–282</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.A. Zhevlakov,   A.M. Slin'ko,   I.P. Shestakov,   A.I. Shirshov,   "Rings that are nearly associative" , Acad. Press  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.I. Zel'manov,   "Jordan division algebras"  ''Algebra and Logic'' , '''18''' :  3  (1979)  pp. 175–190  ''Algebra i Logika'' , '''18''' :  3  (1979)  pp. 286–310</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.A. Skornyakov,   "Elements of general algebra" , Moscow  (1983)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.T. Filippov,   "Central simple Mal'tsev algebras"  ''Algebra and Logic'' , '''15''' :  2  (1976)  pp. 147–151  ''Algebra i Logika'' , '''15''' :  2  (1976)  pp. 235–242</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  I.N. Herstein,   "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR></table>



====Comments====
Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre. For the imbedding problem see [[#References|[a1]]].

The theorem that the only associative division algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568011.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568014.png" />, the algebra of quaternions, is known as Frobenius' theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Skew field constructions" , Cambridge Univ. Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.M. Cohn,   "Algebra" , '''3''' , Wiley  (1991)  pp. Chapt. 7</TD></TR></table>