A Lower Bound for the Class Number of Certain Cubic Number Fields

A Lower Bound for the Class Number of Certain Cubic Number Fields

Let AT be a cyclic number field with generating polynomial i a— 3 ^ û + 3 x3 —Y-x1 -=~-xi and conductor m. We will derive a lower bound for the class number of these fields and list all such fields with prime conductor m = (a1 + 21)/A or m = (1 + 21b2)/A and small class number.

Class-number Problems for Cubic Number Fields

Class number approximation in cubic function fields

A central problem in number theory and algebraic geometry is the determination of the size of the group of rational points on the Jacobian of an algebraic curve over a finite field. This question also has applications to cryptography, since cryptographic systems based on algebraic curves generally require a Jacobian of non-smooth order in order to foil certain types of attacks. There a variety ...

The lower bound for the number of 1-factors in generalized Petersen graphs

‎In this paper‎, ‎we investigate the number of 1-factors of a‎ ‎generalized Petersen graph $P(N,k)$ and get a lower bound for the‎ ‎number of 1-factors of $P(N,k)$ as $k$ is odd‎, ‎which shows that the‎ ‎number of 1-factors of $P(N,k)$ is exponential in this case and‎ ‎confirms a conjecture due to Lovász and Plummer (Ann‎. ‎New York Acad‎. ‎Sci‎. ‎576(2006)‎, ‎no‎. ‎1‎, ‎389-398).

Class number formula for certain imaginary quadratic fields

In this note we shall show how Carlitz in 1954 could have reached an analogue of the Voronoi congruence in the more difficult case of p≡1(mod4): h(-4p) ≡B(p+1)/2(x4)(mod p), where B(p+1)/2(x4) is the generalized Bernoulli number with x4 being the Kronecker symbol associated to the Gaussian field Q(√-4).