Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] lets us decide whether or not a quadratic congruence f ≡ x2 mod π is solvable, where the modulus π is irreducible in F[T ] and f 6≡ 0 mod π. This is similar to the quadratic reciprocity law in Z. We want to develop an analogous reciprocity law when F has characteristic 2. At first it does not seem tha...

In this paper we give an elementary proof of results on the structure of 4-class groups of real quadratic number fields originally due to A. Scholz. In a second (and independent) section we strengthen C. Maire’s result that the 2-class field tower of a real quadratic number field is infinite if its ideal class group has 4-rank ≥ 4, using a technique due to F. Hajir.

2 Class field theory 2 2.1 Number fields and their completions . . . . . . . . . . . . . . . 2 2.1.1 Number fields, prime ideals . . . . . . . . . . . . . . . . 2 2.1.2 Fractional Ideals . . . . . . . . . . . . . . . . . . . . . . 3 2.1.3 Completions . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.4 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . 4 2.1.5 Cycles . . . . . . ....

The Diffie-Hellman key exchange protocol as well as the ElGamal signature scheme are based on exponentiation modulo p for some prime p. Thus the security of these schemes is strongly tied to the difficulty of computing discrete logarithms in the finite field Fp. The Diffie-Hellman protocol has been generalized to other finite groups arising in number theory, and even to the sets of reduced prin...

Matched-field beamforming used in combination with a generalized likelihood ratio test is the most common detector structure in array processing situations. Unfortunately, various array perturbations caused by phase, calibration, propagation effects, or modeling errors can cause the sensor observations to become only partially correlated, limiting the performance of traditional matched-field be...