Let C denote the set of complex numbers, with the usual addition (+) and multiplication (×) operations.
Which of the following are true? If false, indicate what property does not hold.
(C,+) is a group
(C,×) is a group
(C-{0},+) is a group
(C-{0},×) is a group
(C,+,×) is a ring
(C,+,×) is a commutative ring with identity
(C,+,×) is a field
Let Q(√ 2 ) =
{ a + b*√ 2 | a and b are
in Q}. Show that this set, with the usual addition and multiplication, is a field.
Show that √ 3 is not an element of Q(√ 2 ).
Similarly to Q(√ 2 ), we can define
Q(√ 3 ) =
{ a + b*√ 3 | a and b are in Q} and show that it
is a field (you do not have to write our this proof; it is very similar to the proof for
Q(√ 2 ).
Recall that if F1 and F2 are fields, a function f : F1 -> F2 is a field isomorphism if it
is one-to-one and onto, and f(a+b)=f(a)+f(b) and f(ab)=f(a)f(b) for all a and b in F1.
Show that Q(√ 2 ) and
Q(√ 3 ) are not isomorphic as fields (i.e. that there is no field
isomorphism between these two fields).
Hint: assume there exists a field isomorphism f : Q(√ 2 ) ->
Q(√ 3 ), let f(√ 2 ) = x+y*√ 3 .
Consider f(√ 2 *√ 2 ) and derive a contradiction.