I am currently going through a course in Class Field theory and have read that the Artin symbol generalizes the Legendre and Hilbert symbols. I was wondering what field extensions to consider to see this.
I define the Artin and Hilbert maps below for the convenience of users who might not have seen it.
For a cyclic extension $L$ of a number field $K$, define the Artin symbol for an unramified prime $\mathfrak p$, $\displaystyle\left(\frac{L/K}{\mathfrak p}\right)$ to be that element of the Galois group $G$ of $L/K$ that raises an element of $L$ to its $\text{Norm}(\mathfrak p)$-th power. Extend this definition by multiplicativity (in $G$) to all ideals not containing any ramified prime.
For elements $a$ and $b$ in a local field $K$, define the Hilbert symbol $(a,b)$ to be 1 if the equation $z^2 = a x^2 + b y^2$ has a solution $(x,y,z)\in K^3\backslash(0,0,0)$ and -1 otherwise.