The Arithmetic of Quaternion Algebra

1.4. ORDERS AND IDEALS 13
the isotropic groups in H × of the finite subgroup of H 1 are given in the
following table:
group isotropic group
cyclic < sn >, n > 2 < K(sn) × , tn > where tnsn = s −1
n tn, tn ∈ H ×
dicyclic < s2n, j?, n ≤ 2 < s2n, j, 1 + s2n, K × >
binary tetrahedral E24, or < i, j > < K × E24, 1 + i >
binary octahedral E48 K × E48
binary icosahedral E120 K × E120
2. The order of the elements of the finite quaternion group. 1)Prove the elements
in quaternion group of order 4n which are generated by s 2n and j
has the form s t 2nj, where 0 ≤ t ≤ 2n−1 is always of order 4. 2)Find in the
bilinear groups E24, E48, E120 the number of elements with a given order(
it suffices to notice that the elements with reduced trace 0, respectively
,−1, 1, ± √ 2.τor − τ −1 , τ −1 or − τ, are of order 4, respectively, 3, 6, 8, 5,
10). 3)Deduce from 2) that the bilinear octahedral group E48 is not an
isomorphic to the group GL(2, F3) of order 48. 1
3. a characterization of the quaternion fields (Van Praag[1]). prove, if H is
a quaternion field with its center a commutative field K, then the set
consisting of 0 and the elements x ∈ H, x 2 ∈ K, butx /∈ K is an additive
group. Conversely, if H is a field of characteristic different from 2, such
that the set mentioned above is a nonzero additive group, then H is a
quaternion field.
4. The rotations of H (Dieudonné [2] or Bourbaki [3]). A rotation of H is
a proper isometry of the subjacent quaternion space of H. prove every
rotation of H is the mapping of the form:
ua,b : x → axb
, where a, b ar two quaternions such that n(a)n(b) �= 0. Show that two
notations ua,b and uc,d are equal if and only if a = kc, b = k −1 d, where k
is a nonzero element of the center of H (suppose the characteristic is not
2).
1.4 Orders and ideals
The aim of this section is to give some definitions based on the orders, the ideals,
and the reduced discriminants, which will be used in the subsequent chapters,
where K is a local or global field. Our purpose is not to reestablish a frame of
a theory on a Dedekind ring, but only make some definitions more precise, and
these definitions will be adopted in sequel. For a more complete exposition one
can consult the book of Reiner [1] or Deuring [1]. The notations used here are
standard in the following chapters.
Let R be a Dedkind ring, i.e. a noetherian, integrally closed (hence integral)
ring such that all of its nonzero prime ideals are maximal.
Examples: Z, Z[ 1
p ] for prime p, Z[i] and generally the integer ring of a local or
1 This remark is made by Daniel Perrin friendly.