Recent content by shmounal

a) Let an integer $n > 1$ be given, and let $p$ be its smallest prime factor. Show that there can be at most $p − 1$ consecutive positive integers coprime to $n$. b) Show further that the number $p − 1$ in (a) cannot be decreased, by exhibiting $p − 1$ consecutive positive integers coprime $n$...

Thanks for the help! Think I understand now. What I was doing only showed that the degree was 2 or more while finding the min polynomial says it must be exactly 2? i.e. $p(x) = x^2 + 1 \in\mathbb{Q} (\sqrt{2})[x]$ satisfies $p(i) = 0$, ...

Hi, I've been asked to find $[ \mathbb{Q}(\sqrt{2}, i ): \mathbb{Q} ]$, and to write down a basis. Now I know that $[ \mathbb{Q}(\sqrt{2}, i ): \mathbb{Q} ] = 4$, and that a basis could be $ \left \{ 1, i, \sqrt{2}, i\sqrt{2}\right \} $ it is whether the way I am explaining how I arrived here...