T is the set of real numbers of the form a+b√2, where a,b € Q and a,b are not simultaneously zero. Show that (T,•) is a group where "•" is an ordinary multiplication?
1 Answer
For
(1) Closure
(2) Asssociativity
(3) An identity element exists
(3) Inverse elements exist for each member of the set.
Each axiom will be explained and demonstrated in turn
Explanation:
we have
and
To show
(1) Closure:
since
(2) associative property
in this case we have to prove
This is a tedious bit of algebra and it is left as an exercise for the reader to do. But it does work!
(3) identity element exists
so an identity element exists, and is it
(4) Inverse elements exist
where
In this case
take
now since
similarly for
so inverses exist
Since all four axioms are satisfied