Let
.
- Prove that
is a group under multiplication (called the group of roots of unity in
).
- Prove that
is not a group under addition.
- First we show that
is closed under multiplication. Suppose
. Then there exist
such that
. Then
, so that
.
- Multiplication on
is associative since multiplication on
is associative.
- Note that
so that
, and that for all
we have
. So
is an identity in
under multiplication.
- Suppose
with
. Then
, so that
. Moreover,
. So every element of
has a multiplicative inverse in
.
is a group under multiplication.
- First we show that
- Note that
, so that
. However
, and
for all
. So
is not closed under addition and thus not a group.
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