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issisoccer10
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Homework Statement
Prove that there is no isomorphism, [tex]\phi[/tex], from Q under addition to R under addition
Homework Equations
An isomorphism [tex]\phi[/tex]:Q to R is a bijection such that [tex]\phi[/tex](x + y) = [tex]\phi[/tex](x) + [tex]\phi[/tex](y), where x,y are elements of Q
[tex]\phi[/tex](0) = 0.
[tex]\phi[/tex](-x) = -[tex]\phi[/tex](x)
The Attempt at a Solution
My inclination is to attempt to attempt to show a contradiction from two equal rationals p/q and p'/q' occurs when,
[tex]\phi[/tex](p/q - p'/q') = [tex]\phi[/tex](0) = 0, for p,q integers.
So, [tex]\phi[/tex](p/q) - [tex]\phi[/tex](p'/q') = 0,
However, I cannot arrive at an algebraic contradiction.
Is there a better way to go about this proof, relying only on group theory?