Degree

Degree may refer to:

As a unit of measurement

Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field that contains the base field and satisfies additional properties. For instance, the set Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q that includes every real solution to the equation x² = 2.

Definitions

Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under taking inverses in L. In other words, K is a field with respect to the field operations inherited from L. The larger field L is then said to be an extension field of K. To simplify notation and terminology, one says that L / K (read as "L over K") is a field extension to signify that L is an extension field of K.

If L is an extension of F which is in turn an extension of K, then F is said to be an intermediate field (or intermediate extension or subextension) of the field extension L / K.

Degree of an algebraic variety

The degree of an algebraic variety in mathematics is defined, for a projective variety V, by an elementary use of intersection theory.

Definition

For V embedded in a projective space Pⁿ and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined over K, with a linear subspace L in general position, when

Here dim(V) is the dimension of V, and the codimension of L will be equal to that dimension. The degree d is an extrinsic quantity, and not intrinsic as a property of V. For example the projective line has an (essentially unique) embedding of degree n in Pⁿ.

Properties

The degree of a hypersurface F = 0 is the same as the total degree of the homogeneous polynomial F defining it (granted, in case F has repeated factors, that intersection theory is used to count intersections with multiplicity, as in Bézout's theorem).

Other approaches

For a more sophisticated approach, the linear system of divisors defining the embedding of V can be related to the line bundle or invertible sheaf defining the embedding by its space of sections. The tautological line bundle on Pⁿ pulls back to V. The degree determines the first Chern class. The degree can also be computed in the cohomology ring of Pⁿ, or Chow ring, with the class of a hyperplane intersecting the class of V an appropriate number of times.

Erica

Erica/ˈɛrᵻkə/, is a genus of roughly 860 species of flowering plants in the family Ericaceae. The English common names "heath" and "heather" are shared by some closely related genera of similar appearance. The genus Calluna was formerly included in Erica – it differs in having even smaller scale-leaves (less than 2–3 mm long), and the flower corolla being consisting of separate petals. Erica is sometimes referred to as "winter (or spring) heather" to distinguish it from Calluna "summer (or autumn) heather".

Description

Most of the species are small shrubs from 20–150 cm (8–59 in) high, though some are taller; the tallest are E. arborea (tree heath) and E. scoparia (besom heath), both of which can reach up to 7 m (23 ft) tall. All are evergreen, with minute, needle-like leaves 2–15 mm long. Flowers are sometimes axillary, and sometimes borne in terminal umbels or spikes, and are usually outward or downward facing. The seeds are very small, and in some species may survive in the soil for decades.

Erica, Emmen

Erica is a village in the northeastern Netherlands. It is located in Emmen, Drenthe. It has ca. 4800 inhabitants.

Erica began to form in 1863. A fuel shortage prompted large-scale efforts to drain the peats in the area and cut them for fuel. People were drawn to the area and canals were dug, both to drain the peat and to transport the cuts. Buckwheat was farmed on the drained peats. Now, the area's primary industry consists of greenhouse cultivation.

Many of the early inhabitants of Erica were Roman Catholic, and the area is still a Roman Catholic "stronghold".

Coordinates: 52°43′N 6°56′E / 52.717°N 6.933°E / 52.717; 6.933

Erika (given name)

The given name Erika, or Erica, is a feminine form of Eric, deriving from the Old Norse name Eiríkr (or Eríkr in Eastern Scandinavia due to monophthongization). The first element, ei- is derived either from the older Proto-NorseZ*aina(z) meaning "one" or "some", or from Proto-Norse *aiwa(z) meaning "fair" or "tradition". The second element -ríkr derives either from *rík(a)z meaning "ruler" or "prince" (cf. Gothic reiks), or from an even older Proto-Germanic *ríkiaz which meant "powerful" and "rich". The name is thus usually taken to mean something along the lines of "one ruler", "autocrat", "eternal ruler" or "ever powerful".

It is a common name in many Western societies. It is also a popular given name in Japan. Erica is also the name of a genus of approximately 860 species of flowering plants in the family Ericaceae, commonly known as "heaths" or "heathers" in English.

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