Return to Question

Let $\smash{\widehat{\mathbb Z}}^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product topology. For a given prime $p$ let $$ f_p=(1,p)\in\mathbb{Z}_p^\times\ \times\ \prod_{q\ne p}\mathbb{Z}_q^\times. $$ Is the group generated by all $f_p$ dense in $\smash{\widehat{\mathbb{Z}}}^\times$?

Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product topology. For a given prime $p$ let $$ f_p=(1,p)\in\mathbb{Z}_p^\times\ \times\ \prod_{q\ne p}\mathbb{Z}_q^\times. $$ Is the group generated by all $f_p$ dense in $\left(\widehat{\mathbb{Z}}\right)^\times$?

Let $\smash{\widehat{\mathbb Z}}^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product topology. For a given prime $p$ let $$ f_p=(1,p)\in\mathbb{Z}_p^\times\ \times\ \prod_{q\ne p}\mathbb{Z}_q^\times. $$ Is the group generated by all $f_p$ is dense in $\smash{\widehat{\mathbb{Z}}}^\times$?

Let $\smash{\widehat{\mathbb Z}}^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product topology. For a given prime $p$ let $$ f_p=(1,p)\in\mathbb{Z}_p^\times\ \times\ \prod_{q\ne p}\mathbb{Z}_q^\times. $$ Is the group generated by all $f_p$ dense in $\smash{\widehat{\mathbb{Z}}}^\times$?

Let $\widehat{\mathbb Z}^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product topology. For a given prime $p$ let $$ f_p=(1,p)\in\mathbb{Z}_p^\times\ \times\ \prod_{q\ne p}\mathbb{Z}_q^\times. $$ Is the group generated by all $f_p$ is dense in $\widehat{\mathbb{Z}}^\times$?

Let $\smash{\widehat{\mathbb Z}}^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product topology. For a given prime $p$ let $$ f_p=(1,p)\in\mathbb{Z}_p^\times\ \times\ \prod_{q\ne p}\mathbb{Z}_q^\times. $$ Is the group generated by all $f_p$ is dense in $\smash{\widehat{\mathbb{Z}}}^\times$?