*PM : list of all imaginary quadratic extensions whose ring of integers is a PID, id = 9213 new !-- WP guess : list of all imaginary quadratic extensions whose ring of integers is a PID-- Status:
22.
*PM : list of all imaginary quadratic extensions whose ring of integers is a PID, id = 9213 new !-- WP guess : list of all imaginary quadratic extensions whose ring of integers is a PID-- Status:
23.
If the algebraic group is the multiplicative group of a quadratic extension of " N ", the result is the " p " + 1 method; the calculation involves pairs of numbers modulo " N ".
24.
The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if it is assumed to be separable.
25.
In the one-dimensional case, the coefficients form a group of order two, and isomorphism classes of twisted forms of "'G "'m are in natural bijection with separable quadratic extensions of " K ".
26.
They used additional structures : in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they use elliptic curves with complex multiplication and their points of finite order.
27.
One can give an effective description of the set of such curves in an arithmetic surface or three manifold : they correspond to certain units in certain quadratic extensions of the base field ( the description is lengthy and shall not be given in full here ).
28.
Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field " F " containing two points on the line, the center of the circle, and the radius of the circle.
29.
The rupture field of X ^ 2 + 1 over \ mathbb F _ 3 is \ mathbb F _ 9 since there is no element of \ mathbb F _ 3 with square equal to-1 ( and all quadratic extensions of \ mathbb F _ 3 are isomorphic to \ mathbb F _ 9 ).
30.
If one adjoins the Vandermonde polynomial to the ring of symmetric polynomials in " n " variables \ Lambda _ n, one obtains the quadratic extension \ Lambda _ n [ V _ n ] / \ langle V _ n ^ 2-\ Delta \ rangle, which is the ring of alternating polynomials.
