Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic...

In mathematics, there are several theorems basic to algebraic K-theory. Throughout, for simplicity, we assume when an exact category is a subcategory of...

In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to R [ t ] {\displaystyle...

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations...

a root, where k is chosen so that deg(f) + 2k ∈ I). Another algebraic proof of the fundamental theorem can be given using Galois theory. It suffices to...

In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those...

In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the...

Therefore, these two theorems are equivalent. There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant...

statement in-and-of itself. Fundamental theorem of algebra Fundamental theorem of algebraic K-theory Fundamental theorem of arithmetic Fundamental theorem of...

the basic theorems in algebraic number theory are the going up and going down theorems, which describe the behavior of some prime ideal p ∈ Spec ( O K )...

incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results...

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known...

cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the...

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants...

19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations...

algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and...

the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups, including...

In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations...

studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its...

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician...

groups. It was proved by Évariste Galois in his development of Galois theory. In its most basic form, the theorem asserts that given a field extension E/F...

theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is...

describe rational homotopy theory in algebraic terms. The definition of a Lie algebra over a field extends to define a Lie algebra over any commutative ring...

or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra. Replacing the field of scalars by...

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz...

are discussed in the article on distributivity in order theory. Some additional order structures that are often specified via algebraic operations and...

field K [ T ] {\displaystyle K[T]} . A matroid that can be generated in this way is called an algebraic matroid. No good characterization of algebraic matroids...

noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: Generic matrix ring. A homomorphism between two R-algebras is an...

juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives...

point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional...