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Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field. [16] The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. [16] At this point, he already knew the relationship between the discriminant and ...
A place of a number field is an equivalence class of absolute values on [6] pg 9. Essentially, an absolute value is a notion to measure the size of elements of . Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity).
The 3-tuple number set (,,) denotes radial distance, the polar angle—"inclination", or as the alternative, "elevation"—and the azimuthal angle. It is the common practice within the physics convention, as specified by ISO standard 80000-2:2019 , and earlier in ISO 31-11 (1992).
The discriminant of N is d 3 f 4. [6] [7] The field K is a pure cubic field if and only if d = −3. This is the case for which the quadratic field contained in the Galois closure of K is the cyclotomic field of cube roots of unity. [7]
Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units . [ 1 ]
A field is called a prime field if it has no proper (i.e., strictly smaller) subfields. Any field F contains a prime field. If the characteristic of F is p (a prime number), the prime field is isomorphic to the finite field F p introduced below. Otherwise the prime field is isomorphic to Q. [14]
Note that if K is Galois over then either r 1 = 0 or r 2 = 0.. Other ways of determining r 1 and r 2 are . use the primitive element theorem to write = (), and then r 1 is the number of conjugates of α that are real, 2r 2 the number that are complex; in other words, if f is the minimal polynomial of α over , then r 1 is the number of real roots and 2r 2 is the number of non-real complex ...
A node of a point quadtree is similar to a node of a binary tree, with the major difference being that it has four pointers (one for each quadrant) instead of two ("left" and "right") as in an ordinary binary tree. Also a key is usually decomposed into two parts, referring to x and y coordinates.