Polar Form
Let Gr and Gθ be subgroups of the multiplicative group of the nonzero elements of GI(p), of orders (p − 1)/2 and 2(p + 1), respectively. Then all nonzero elements of GI(p) can be written in the form ζ = α·β, where α ∈ Gr and β ∈ Gθ.
Considering that any element of a cyclic group can be written as an integral power of a group generator, it is possible to set r = α and εθ = β, where ε is a generator of . The powers εθ of this element play the role of ejθ over the complex field. Thus, the polar representation assumes the desired form, where r plays the role of the modulus of ζ. Therefore, it is necessary to define formally the modulus of an element in a finite field. Considering the nonzero elements of GF(p), it is a well-known fact that half of them are quadratic residues of p. The other half, those that do not possess square root, are the quadratic non-residue (in the field of real numbers, the elements are divided into positive and negative numbers, which are, respectively, those that possess and do not possess a square root).
The standard modulus operation (absolute value) in always gives a positive result.
By analogy, the modulus operation in GF(p) is such that it always results in a quadratic residue of p.
The modulus of an element, where p = 4k + 3, is
The modulus of an element of GF(p) is a quadratic residue of p.
The modulus of an element a + jb ∈ GI(p), where p = 4k + 3, is
In the continuum, such expression reduces to the usual norm of a complex number, since both, a2 + b2 and the square root operation, produce only nonnegative numbers.
- The group of modulus of GI(p), denoted by Gr, is the subgroup of order (p − 1)/2 of GI(p).
- The group of phases of GI(p), denoted by G, is the subgroup of order 2(p + 1) of GI(p).
An expression for the phase as a function of a and b can be found by normalising the element (that is, calculating ), and then solving the discrete logarithm problem of /r in the base over GF(p). Thus, the conversion rectangular to polar form is possible.
The similarity with the trigonometry over the field of real numbers is now evident: the modulus belongs to GF(p) (the modulus is a real number) and is a quadratic residue (a positive number), and the exponential component ) has modulus one and belongs to GI(p) (e also has modulus one and belongs to the complex field ).
Read more about this topic: Trigonometry In Galois Fields
Famous quotes containing the words polar and/or form:
“Professor Fate: My apologies. There’s a polar bear in our car.”
—Arthur Ross. Professor Fate (Jack Lemmon)
“It is a conquest when we can lift ourselves above the annoyances of circumstances over which we have no control; but it is a greater victory when we can make those circumstances our helpers,—when we can appreciate the good there is in them. It has often seemed to me as if Life stood beside me, looking me in the face, and saying, “Child, you must learn to like me in the form in which you see me, before I can offer myself to you in any other aspect.””
—Lucy Larcom (1824–1893)