Homework 8

Subfields of the field of real numbers

Let Q denote the set of rational numbers.

• 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. Consider the function f : Q(√ 2 ) -> Q(√ 3 ) given by f(a+b*√ 2 ) = a+b*√ 3 . Show that f is not a field isomorphism.
• 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.