I remember a friend in graduate school throwing an early edition of Jurgen Neukirch's Algebraic Number Theory book against a wall (so hard that it split the binding) after he had worked for a number of days to reconcile something he realized was an error (or typo) in the book.


For the life of me, I cannot remember what this error was. 



> 
> Has anyone come across a particularly
>  annoying error in an early edition of
>  Neukirch? If so, what was it?
> 
> 
> 




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To give context to some of the answers below, I include the original statement of this question:



> 
> What are your "favorite" annoying errors and typos in otherwise excellent sources?
> 
> 
> 


Please especially try to include subtle errors that can mislead, as these are very important.


Disclaimer: The author of this quesion thinks that working to resolve an error like this is a very worthwhile exercise, at least from the standpoint of having cool war stories that make a topic memorable. I'm not sure, therefore, whether I would want to post an answer to this question myself. I posted the question because I thought it would be fun.



Has anyone found an error in an early version of Neukirch?
In my edition of Neukirch, Chapter I.9, Exercise 2:


If $L|K$ is a Galois extension of algebraic number fields, and $\mathfrak{P}$ a prime ideal which is unramified over $K$ (i.e. $\mathfrak{p} = \mathfrak{P} \cap K$ is unramified in $L$), then there is one and only one automorphism $\phi\_{\mathfrak{P}} \in G(L|K)$ such that


$\phi\_{\mathfrak{P}}a \simeq a^q \ mod \ \mathfrak{P}$ for all $a \in \mathcal{O}$,


where $q = [\kappa(\mathfrak{P}) : \kappa(\mathfrak{p})]$. It is called the Frobenius automorphism. The decomposition group $G\_{\mathfrak{P}}$ is cyclic and $\phi\_{\mathfrak{p}}$ is a generator of $G\_{\mathfrak{P}}$.


Typo: That should be $q = |\kappa(\mathfrak{p})|.$


