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Let $G=\mathbb{Q}^{x}$ be the multiplicative group of nonzero rational numbers. If $\alpha=p / q \in G$, where $p$ and $q$ are relatively prime integers, let $\varp...

Question

Let $G=\mathbb{Q}^{x}$ be the multiplicative group of nonzero rational numbers. If $\alpha=p / q \in G$, where $p$ and $q$ are relatively prime integers, let $\varphi: G \rightarrow G$ be the map which interchanges the primes 2 and 3 in the prime power factorizations of $p$ and $q$ (so, for example, $\varphi\left(2^{4} 3^{11} 5^{1} 13^{2}\right)=3^{4} 2^{11} 5^{1} 13^{2}, \varphi(3 / 16)=\varphi\left(3 / 2^{4}\right)=2 / 3^{4}=2 / 81$, and $\varphi$ is the identity on all rational numbers with n

Let $G=\mathbb{Q}^{x}$ be the multiplicative group of nonzero rational numbers. If $\alpha=p / q \in G$, where $p$ and $q$ are relatively prime integers, let $\varphi: G \rightarrow G$ be the map which interchanges the primes 2 and 3 in the prime power factorizations of $p$ and $q$ (so, for example, $\varphi\left(2^{4} 3^{11} 5^{1} 13^{2}\right)=3^{4} 2^{11} 5^{1} 13^{2}, \varphi(3 / 16)=\varphi\left(3 / 2^{4}\right)=2 / 3^{4}=2 / 81$, and $\varphi$ is the identity on all rational numbers with numerators and denominators relatively prime to 2 and to 3 ). (a) Prove that $\varphi$ is a group isomorphism. (b) Prove that there are infinitely many isomorphisms of the group $G$ to itself. (c) Prove that none of the isomorphisms above can be extended to an isomorphism of the ring $\mathbb{Q}$ to itself. In fact prove that the identity map is the only ring isomorphism of $\mathbb{Q}$.



Answers

In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation $$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0}=0$$ and let $\frac{P}{2}$ be a rational root reduced to lowest terms. a. Substitute $\frac{p}{q}$ for $x$ in the equation and show that the equation can be written as $$a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}=-a_{0} q^{n}$$ b. Why is $p$ a factor of the left side of the equation? c. Because $p$ divides the left side, it must also divide the right side. However, because $\frac{P}{q}$ is reduced to lowest terms, $p$ and $q$ have no common factors other than $-1$ and 1 Because $p$ does divide the right side and has no factors in common with $q^{n},$ what can you conclude? d. Rewrite the equation from part (a) with all terms containing $q$ on the left and the term that does not have a factor of $q$ on the right. Use an argument that parallels parts (b) and (c) to conclude that $q$ is a factor of $a_{n}$.


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