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A variable or value of that type is usually represented as a fraction m/n where m and n are two integer numbers, either with a fixed or arbitrary precision.Depending on the language, the denominator n may be constrained to be non-zero, and the two numbers may be kept in reduced form (without any common divisors except 1).
every element x of GF(2) satisfies x 2 = x (i.e. is idempotent with respect to multiplication); this is an instance of Fermat's little theorem. GF(2) is the only field with this property (Proof: if x 2 = x, then either x = 0 or x ≠ 0. In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives x = 1 ...
Then, f(r) = 0, which can be rearranged to express r k as a linear combination of powers of r less than k. This equation can be used to reduce away any powers of r with exponent e ≥ k. For example, if f(x) = x 2 + 1 and r is the imaginary unit i, then i 2 + 1 = 0, or i 2 = −1. This allows us to define the complex product:
The vertex set of H is the Cartesian product V(G 1) × V(G 2), where V(G 1) and V(G 2) are the vertex sets of G 1 and G 2, respectively. Two vertices (a 1,a 2) and (b 1,b 2) of H are connected by an edge, iff a condition about a 1, b 1 in G 1 and a 2, b 2 in G 2 is fulfilled. The graph products differ in what exactly this condition is.
Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday non-mathematical or non-programming circumstances.
Using the XOR swap algorithm to exchange nibbles between variables without the use of temporary storage. In computer programming, the exclusive or swap (sometimes shortened to XOR swap) is an algorithm that uses the exclusive or bitwise operation to swap the values of two variables without using the temporary variable which is normally required.
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On the other hand, the maximal real subfields Q(cos(2π/2 n)) of the 2-power cyclotomic fields Q(ζ 2 n) (where n is a positive integer) are known to have class number 1 for n≤8, [8] and it is conjectured that they have class number 1 for all n. Weber showed that these fields have odd class number.