page_title: Field (mathematics)
section_title: Division rings
hierarchical_section_title: Field (mathematics) / Related notions / Division rings
caption_attribution_description: English: A failed attempt to comb a hairy ball flat. (I, anonymous author of this work, claim no author's rights.)
context_page_description: In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
  The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
  The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable.
context_section_description: Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all axioms of fields, except for multiplicative inverses. Dropping instead the condition that multiplication is commutative leads to the concept of a division ring or skew field. The only division rings that are finite-dimensional R-vector spaces are R itself, C (which is a field), the quaternions H (in which multiplication is non-commutative), and the octonions O (in which multiplication is neither commutative nor associative). This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor. The non-existence of an odd-dimensional division algebra is more classical. It can be deduced from the hairy ball theorem illustrated at the right.
caption: The hairy ball theorem states that a ball cannot be combed. More formally, there is no continuous tangent vector field on the sphere S2, which is everywhere non-zero.
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