InfoAlgebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology.

The field of rational numbers Q has been introduced above.A related class of fields very important in number theory are algebraic number fields.…

Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.The collection of all algebraic objects of a given type will usually be a proper class.Examples include the class of all groups, the class of all vector spaces, and many others.…

For example, in the quadratic polynomial the 3 is a constant term.After like terms are combined, an algebraic expression will have at most one constant term.Thus, it is common to speak of the quadratic polynomial where x is the variable, and has a constant term of c. If c = 0, then the constant term will not actually appear when the quadratic is written.…

In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action instead of a space with group action, as in stable homotopy theory.The field has become more active recently because of its connection to algebraic K-theory.…

One can perform algebraic operations on power series to generate new power series.Besides the ring structure operations defined above, we have the following.…

During this time he did much important work outside of his work on algebraic geometry.

If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will not be totally real, although it is a field of real numbers.The totally real number fields play a significant special role in algebraic number theory.An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two.…

Richard Gordon Swan is an American mathematician who is known for Swan's theorem and the Swan representation.His work has mainly been in the area of algebraic K-theory.…

But if R is in fact a ring of algebraic integers, then the class number is always finite.This is one of the main results of classical algebraic number theory.Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraic number field of small discriminant, using Minkowski's bound.…

In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.As an algebraic structure, every field is a ring, but not every ring is a field.The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example the integers form a ring, but 2x = 1 has no solution in integers.…

Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.Algebraic number theory is a special case of group theory, thereby following the rules of the latter.…

Fields of algebraic numbers are also called algebraic number fields, or shortly number fields.

This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.The most common application of the plus construction is in algebraic K-theory.…

An example of where this is used is in algebraic number theory in the theory of the different ideal.The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K. The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.…

For example, Q is a global field, and the attached local fields are Q p and R (Ostrowski's theorem).Algebraic number fields and function fields over F q are further global fields.Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally -- this technique is called local-global principle.…

The plus sign can also indicate many other operations, depending on the mathematical system under consideration.Many algebraic structures have some operation which is called, or equivalent to, addition.It is conventional to use the plus sign to only denote commutative operations.…

In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined.The work is now considered the foundation stone and basic reference of modern algebraic geometry.Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published.…

Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology.The added structure must be compatible, in some sense, with the algebraic structure.Algebraic structures are defined through different configurations of axioms.…

Note that the sequence of groups and homomorphisms may be either finite or infinite.A similar definition can be made for certain other algebraic structures.For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms.…

These noncommutative algebras, and the non-associative Lie algebras, were studied within universal algebra before the subject was divided into particular mathematical structure types.One sign of re-organization was the use of direct sums to describe algebraic structure.The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928).…