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The algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective. If K is a function field over k of n variables, and L is a function field in m variables, and n > m, then there are no morphisms from K to L.
The function field of the affine line over K is isomorphic to the field K(t) of rational functions in one variable. This is also the function field of the projective line . Consider the affine algebraic plane curve defined by the equation y 2 = x 5 + 1 {\displaystyle y^{2}=x^{5}+1} .
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. [1] [2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.
The function field of X is the same as the one of any open dense subvariety. In other words, the function field is insensitive to replacing X by a (slightly) smaller subvariety. The function field is invariant under isomorphism and birational equivalence of varieties.
A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions ...
The Langlands conjectures for function fields state (very roughly) that there is a bijection between cuspidal automorphic representations of GL n and certain representations of a Galois group. Drinfeld used Drinfeld modules to prove some special cases of the Langlands conjectures, and later proved the full Langlands conjectures for GL 2 by ...
Examples include the map from a wedge of ... The fact that this projection is a branched covering of degree n may also be seen by considering the function fields.
For example, the dimension of an algebraic variety is the transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory in positive characteristic a role that is very similar to the role of algebraic number fields in characteristic zero.