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.Q:

Existence of isomorphism between two complex numbers

Let $z_1,z_2\in\mathbb{C}$ be two complex numbers that satisfy $|z_1|=|z_2|=1$. Prove that there is a isomorphism of $\mathbb{C}$ to itself that maps $z_1$ to $z_2$ and vice versa.

For the proof I tried to denote $$f(z)=z+az^3$$ where $a=\bar{z_1}=\overline{\bar{z_2}}$. Hence I tried to prove by contradiction that $f$ is a surjective map of $\mathbb{C}$ to itself such that $f(z_1)=z_2$:
$z_1\mapsto z_2\Rightarrow f(z_1)=z_2$ and $f(z_1)
e z_2$ since $z_1
otin\text{im}(f)$
Hence there is a unique element $z_0$ such that $f(z_0)=z_1$. But how can I continue from here?

A:

If $z_1$ and $z_2$ are complex conjugates of each other, then $|z_1|=1$ implies that $z_1=\bar z_1$ and thus $f(z_1)=z_2$. Now the opposite direction is also true, so you have a bijective map from $z_1$ to $z_2$ and vice versa.

Jill Ireland

Jill Ireland (born June 16, 1964) is a long-distance runner from Canada.

References

External links

Category:1964 births
Category:Living people
Category:Canadian female long-distance runners
Category:Canadian female marathon runners
Category:Canadian masters athletes
Category:World Athletics Championships athletes for Canada
Category:Athletes (track and field) at the 1991 Pan American Games
Category:Pan American Games medalists in athletics (track and field)
Category:Pan American Games silver medalists for Canada
Category:Athletes (track and field) at the 1992 Summer Olympics
Category:Olympic track and field

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