Difference between the Artin symbol and the Frobenius element? The title says it all really! 
I have been reading 'Primes of the form $x^2+ny^2$', by Cox, and in chapter 5 he introduces the Artin Symbol, which for a field extension, $L/K$ is the unique element, $\sigma\in$ Gal($L/K$) that satisfies $\sigma(\alpha)\equiv\alpha^{N(\mathfrak{p})}\pmod{\mathfrak{P}}$, where $\mathfrak{P}$ is a prime of $\mathcal{O}_L$ containing $\mathfrak{p}$.
Similarly, in J.Milne's class field theory notes (http://www.jmilne.org/math/CourseNotes/CFT.pdf) he defines the frobenius element as the unique element in Gal($L/K$) satisfying:


*

*The same condition as above

*$\sigma\in D_{\mathfrak{P}}$, the decomposition group of $\mathfrak{P}|\mathfrak{p}$.


I was just wondering if, in general use, there is a huge difference between these two, or if we refer to one of them in a certain case, but the other one in a different situation, and why there are two names for it if they are the same thing. 
 A: They are the same thing. Cox's book also has $\sigma \in D_\mathfrak{P}$ stated somewhere. Note that $\mathfrak{p}$ has to be unramified in $L$ for this definition to make sense.
The one slight difference in use is that the term "the Artin symbol" means exactly this situation, but "the Frobenius element" is a little bit ambiguous. Without any qualifications, "the Frobenius" is the generator of the (cyclic) Galois group of a finite extension of finite fields. It doesn't have to come from an extension of number fields.
If it does in fact come from the localization and reduction modulo $\mathfrak{p}$ of a Galois extension $L/K$ of number fields, then it's possible to pull the Frobenius element back to $Gal(L/K)$. But then it's not unique unless $Gal(L/K)$ is abelian. The ambiguity is in the choice of a prime $\mathfrak{P}$ over $\mathfrak{p}$, hence the term "Frobenius at $\mathfrak{P}$". When the prime is not unramified, you get an element in $D_\mathfrak{P}/I_\mathfrak{P}$ instead. 
The point is that "the Frobenius" could make sense when there are no number fields around, but "the Artin symbol" refers to a specific Frobenius element in this particular set-up where you have an extension of number fields, in the context of class field theory. 
