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A norm on the integral Hamiltonians

Let be the ring of integral Hamiltonian Quaternions and define by

.

1. Prove that for all , where

.

2. Prove that for all .

3. Prove that an element is a unit if and only if it has . Then

show that . [Hint: The inverse of in the rational quaternions is

.]

Let and .

1.

, after a bunch

of cancellation.

2.

. We have

omitted a lengthy cancellation step.

3. Suppose first that is a unit. Then for some , where is an integral

Hamiltonian Quaternion. Note from the definition of as a sum of squares

that for all . Now , and

and are both integers. Thus . Now suppose

. Then , and clearly , so that is a unit in .

Suppose is a unit. Then . If any of the

is at least 2 in absolute value, we have a contradiction. Thus each

is either 0,1, or -1. If more than one has absolute value 1, we have

another contradiction; thus at most one of is 1. Clearly if none are

zero, then \alpha = 0$ is not a unit, and if exactly one is 1, then , so

that is a unit. Thus . Since is nonabelian and has six elements

of order 4, .

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computationconjugate

counterexample cycle

notation cyclic group degree

dihedral group directproduct direct sum divisibility

factorization field fieldextension finite field

finite group gaussian

integers general linear group

generating set greatest

common divisor groupgroup action group

homomorphism group

presentation Gröbner basis idealintegers integral domain

intersection irreducible

polynomial isomorphism jordan

canonical form kernel linear

transformation matrix maximal

ideal modular arithmetic

module norm normalizer

normal subgrouporder (group element)polynomial polynomialring prime prime ideal

principal ideal principal ideal

domain quadratic f ield quadratic

integer ring quaternion group

quotient group quotient ring

rational numbers rationals

real numbers relatively prime

ring semidirect product

semigroup simple group

subgroup subgroup index

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