Cyclic Quartic Fields Research Articles

Cyclic quartic fields and genus theory of their subfields

Let k = Q (√ u) ( u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = Q (√ vwη), where η is fixed in k and satisfies η ⪢ 1, (η) = U 2√ u, | U 2| = |(√ u)|, ( v, u) = 1, v ∈ Z is squarefree, w| u, 0 < w < √ u. Thus if u ≠ a 2 + b 2, there is no K ⊃ k. If u = a 2 + b 2 then for each fixed v there are 2 g − 1 K ⊃ k, where g is the number of prime divisors of u. (2) K k has a relative integral basis (RIB) (i.e., O K is free over O k ) iff N( ε 0) = −1 and w = 1, where ε 0 is the fundamental unit of k, (or, equivalently, iff K = Q (√ vε 0√ u), ( v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc( K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that K k has a RIB if u is a prime; (ii) Edgar and Peterson ( J. Number Theory 12 (1980), 77–83) showed that for u composite there is at least one K ⊃ k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert ( Ann. of Math. 31 (1930), 381–418) are wrong.

Products and sums of powers of binomial coefficients mod 𝑝 and solutions of certain quaternary Diophantine systems

In this paper we prove that certain products and sums of powers of binomial coefficients modulo p = q f + 1 p = qf + 1 , q = a 2 + b 2 q = {a^2} + {b^2} , are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition \[ 16 p α = x 2 + 2 q u 2 + 2 q v 2 + q w 2 , ( x , u , v , w , p ) = 1 , x w = a v 2 − 2 b u v − a u 2 , x ≡ 4 ( mod q ) , α ⩾ 1. \begin {array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q{v^2} + q{w^2},\quad (x,u,v,w,p) = 1,} \\ {xw = a{v^2} - 2buv - a{u^2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end {array} \] The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field \[ K = Q ( i 2 q + 2 a q ) , K = Q\left ( {i\sqrt {2q + 2a\sqrt q } } \right ), \] as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field Q ( e 2 π i p / q ) Q({e^{2\pi ip/q}}) . Let the four cosets of the subgroup A of quartic residues be given by c j = 2 j A , j = 0 , 1 , 2 , 3 {c_j} = {2^j}A,j = 0,1,2,3 , and let \[ s j = 1 q ∑ t ∈ c j t , j = 0 , 1 , 2 , 3. {s_j} = \frac {1}{q}\sum \limits _{t \in {c_j}} {t,\quad j = 0,1,2,3.} \] Let s m {s_m} and s n {s_n} denote the smallest and next smallest of the s j {s_j} respectively. We give new, and unexpectedly simple determinations of Π k ∈ c n k f ! {\Pi _{k \in {c_n}}}kf! and Π k ∈ c n + 2 k f ! {\Pi _{k \in {c_{n + 2}}}}kf! , in terms of the parameters x in the above partition of 16 p α 16{p^\alpha } , in the complicated case that arises when the class number of K is &gt; 1 &gt; 1 and s m ≠ s n {s_m} \ne {s_n} .

On class numbers of imaginary quadratic and quartic fields

1. Introduction. It is known (cf. [1], [3], [4], [5], [7]) that there exist infinitely many imaginary quadratic fields each with class number divisible by a given integer. In this paper we show another simple proof of the above theorem. By a similar method we also construct infinitely many imaginary cyclic quartic fields whose relative class numbers are divisible by l, where l is a given prime congruent to 1 modulo 4. Further we characterize the imaginary quadratic fields whose class numbers are multiples of a given prime, by describing them explicitly. In the following, for an integer :r in an algebraic number field K we denote by (~) the principal ideal in K generated by ~. We write p ]] n for a prime p and an