May 29th, 2019, 01:56 PM  #1 
Newbie Joined: Jul 2018 From: morocco Posts: 26 Thanks: 0 Math Focus: algebraic number theory  exercise
Hello Let $p$ be a prime such that $p\equiv 1\pmod 8$ and $K=\mathbb Q(\sqrt[4]{2})$. How to show that : $p$ decomposes into $4$ primes of $K$ if and only if $\left(\frac{2}{ p}\right)_4=1$. thanks 
May 29th, 2019, 06:11 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 670 Thanks: 440 Math Focus: Dynamical systems, analytic function theory, numerics 
This doesn't make sense. Your notation for $K$ seems to imply that $K$ is a field extension. However, fields do not contain primes so I'm not sure what you mean by the last line.

May 30th, 2019, 02:52 AM  #3 
Newbie Joined: Jul 2018 From: morocco Posts: 26 Thanks: 0 Math Focus: algebraic number theory 
Here $K$ is a number field, so we mean by a prime of $K$, a prime of the ring of integers of $K$.


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