# Skew-field

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A skew-field (or skew field) is a ring in which the equations $ax=b$ and $ya=b$ with $a\ne 0$ are uniquely solvable. In the case of an associative ring (cf. Associative rings and algebras) it is sufficient to require the existence of a unit 1 and the unique solvability of the equations $ax=1$ and $ya=1$ for any $a\ne 0$. A commutative associative skew-field is called a 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
$$\begin{pmatrix}a & \bar b\\ -b & \bar a\end{pmatrix}$$ over the field of complex numbers with the usual operations (see Quaternion). An example of a non-associative skew-field is the 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). Any skew-field is a division algebra either over the field of rational numbers or over a field of residues $\F_p = \Z/(p)$. The skew-field of quaternions is a $4$-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is $8$-dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [Ad], and also 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 [Po]). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see [Sk], [He]). 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 [ZhSlShSh]. A similar result applies to Mal'tsev skew-fields [Fi] (see Mal'tsev algebra) and to Jordan skew-fields [Ze] (see 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).
The theorem that the only associative division algebras over $\R$ are $\R$, $\C$ or $\mathbb{H}$, the algebra of quaternions, is known as Frobenius' theorem.