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Each subfield of the field of complex numbers contains every rational number


Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.7

Solution: Every subfield of $\C$ has characterisitc zero since if $\mb F$ is a subfield then $1\in \mb F$ and $n\cdot 1=0$ in $\mb F$ implies $n\cdot1=0$ in $\C$. But we know $n\cdot1=0$ in $\C$ implies $n=0$. So $1,2,3,\dots$ are all distinct elements of $\mb F$. And since $\mb F$ has additive inverses $-1,-2,-3,\dots$ are also in $\mb F$. And since $\mb F$ is a field also $0\in \mb F$. Thus $\Z\subseteq \mb F$.

Now $\mb F$ has multiplicative inverses so $\pm\frac1n\in \mb F$ for all natural numbers $n$. Now let $\frac mn$ be any element of $\mb Q$. Then we have shown that $m$ and $\frac1n$ are in $\mb F$. Thus their product $m\cdot\frac1n$ is in $\mb F$. Thus $\frac mn\in \mb F$. Thus we have shown all elements of $\mb Q$ are in $\mb F$.


Linearity

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