Does the Legendre Symbol/quadratic reciprocity generalize to higher degrees? The Legendre symbol is a tool for measuring whether or not
$$
x^2 \equiv a \text{ } (p)
$$
has a solution in $\mathbb{F}_p$ for some fixed integer $a$. Does the Legendre symbol generalize to higher degrees? For example, can I define a law
$$
\left( \frac{\cdot}{p} \right)_k:\mathbb{F}_p^* \to ???
$$
telling me whether or not
$$
\frac{\mathbb{F}_p[x]}{(x^k - a)}
$$
is a field, and if it is not, how far is it from being a field? Also, if there is such a rule, are there reciprocity laws which can be found?
 A: Your question about  a  generalization of the Legendre symbol/the quadratic reciprocity law was known as Hillbert’s 9th problem and is solved completely by class field theory , CFT for short. Here is a quick – but necessarily not short ! - glimpse at that vast magnificent landscape (for a history of CFT, see e.g. Cassels-Fröhlich, chapter XI):
Given a number field $K$, CFT aims to « describe » the abelian extensions $L/K$ in terms of the « arithmetic » of $K$ alone. This formulation is rather vague, but let us start from the quadratic reciprocity law. The initial problem was the study of the congruence $x^2  - a \equiv 0 $ mod $p$, where $p$ is a prime not dividing $a$. Gauss’ reciprocity means that the existence of a solution depends only on the arithmetic progression mod $4a$ to which $p$ belongs. But a well known result says that, for a quadratic field $L=\mathbf Q (\sqrt d)$, an odd prime $p$ splits completely in $L$ iff $p$ does not divide $d$ and $d$ is a square mod $p$ (this is the second part of your question).The splitting of the prime $p$ is the link with the aforementioned goal of CFT, because of the following classical theorem : let $L_i,  i = 1, 2$ be two Galois extensions (not necessarily abelian) extensions of $K$, and let $ Spl(L_{i}/K)$ be the set of primes of $K$ which split completely in $L_i$ ; then $L_1 = L_2$ iff  $ Spl(L_{1}/K) = Spl(L_{2}/K) $. Thus CFT for quadratic number fields, in view of Gauss’ reciprocity, can be expressed in terms of the arithmetic of $\mathbf Q$, more precisely in terms of congruences.
If we stick  to the base field $\mathbf Q$, CFT in the above sense can actually be derived entirely from the Kronecker-Weber theorem, which asserts that any abelian extension $L$ of $\mathbf Q$ is contained in a cyclotomic field $\mathbf Q(\zeta_m)$. The integer $m$ is called a defining modulus for $L$, and the smallest such modulus, denoted $f_L$, is the conductor of $L$. For $a \in \mathbf Z$, coprime to $m$, the Artin symbol ($L/a$) is defined as the restriction to $L$ of the automorphism of $C_m := (\mathbf Z/m\mathbf Z)^* := Gal(\mathbf Q(\zeta_m)/ \mathbf Q)$ which sends $\zeta_m$ to $\zeta_{m}^{a}$. The symbol $(L/.)$ is a generalization of the Legendre and Jacobi symbols, and it gives an exact sequence $1 \to I_{L,m} \to C_m \to Gal(L/\mathbf Q) \to 1$, where the kernel $ I_{L,m}$ is defined tautologically. This sequence, usually called the Artin reciprocity law,  generalizes Gauss’ reciprocity. A prime $p$ splits completely in $L$ iff  $p$ represents a class of $ I_{L,m}$  mod $f_L$ .
To get genuine « higher » symbols and reciprocity, you have to start from a number field $K \neq \mathbf Q$, but then comes the very hard part of CFT. Let us only say that, given an abelian $L/K$,  the notion of a defining modulus $\mathcal M$ for $L/K$ can be adequately generalized (here, if one « neglects » the infinite primes,  $\mathcal M$ is an ideal of the ring of integers $A_K$ of $K$), $C_m$ is replaced by $C_{\mathcal M}$ := $A_\mathcal M / R_{\mathcal M}$, where $A_\mathcal M$ is the group of fractional ideals coprime to $\mathcal M$ and $R_{\mathcal M}$ is the ray subgroup mod $\mathcal M$,  the Artin symbol $(L/K,.)$ is defined by $(L/K, \mathcal P) (\alpha) \equiv \alpha^{N\mathcal P}$ mod $\mathcal P . A_L$ (here  $N\mathcal P$ is the norm  of the ideal $\mathcal P$ of $A_K$), and the Artin reciprocity law  is the exact sequence  $1 \to I_{L/K, \mathcal M} := N_{L/K} (C_{L,\mathcal M}) \to C_{K,\mathcal M} \to Gal(L/K) \to 1$
Concerning specifically your question, let me refer only to the exercises 1 and 2 of Cassels-Fröhlich. For a number field $K$ containing the group of $m$-th roots of 1, an $m$-th power residue symbol is defined in ex. 1.1, and is shown to give a direct extension of the Legendre symbol/quadratic residue (which correspond to $K=\mathbf Q$ and $m=2$). Similarly, the Hilbert symbols are defined in ex. 2, and the product formula proved in ex. 2.9 also generalizes directly the quadratic reciprocity law ./.
