Let $p$ be a prime $p \equiv 1 \bmod 4$. An $x\in\mathbb{Z}/p\mathbb{Z}$ is a quartic residue if there exists a $y$ such that $x = y^4 \bmod p$. Like for quadratic residues, there is a symbol which outputs whether a quadratic residue $a$ is a quartic residue modulo $p$, defined as the symbol $\left(\frac{a}{p}\right)_4$ satisfying $$\left(\frac{a}{p}\right)_4 = a^{(p-1)/4}\bmod p$$ Then $a$ is a quartic residue iff $\left(\frac{a}{p}\right)_4 = 1$.
My question is: can the quartic residue symbol be defined for prime powers? I would expect $$\left(\frac{a}{p^k}\right)_4 = \left(\left(\frac{a}{p}\right)_4\right)^k$$ after the Jacobi symbol. Any references on such an extension of the quartic residue symbol would also be much appreciated.
This is a partial answer because there are two related ways that your question can be answered: we can ask when $a$ is a quartic residue modulo $p^k$, or we can ask if the Jacobi-like symbol that you defined has any special properties. I only have something to say about the first of these questions.
If you want to define the quartic residue symbol in such a way that $$ \left( \frac{a}{p^k} \right)_4 $$ is equal to $1$ if and only if $a$ is a quartic residue modulo $p^k$ where $p$ is an odd prime, and $a$ is not divisible by $p$, then it turns out that $$ \left( \frac{a}{p^k} \right)_4 = \left( \frac{a}{p} \right)_4. $$
This is a special case of Hensel's Lemma.
It's clear that if $a$ is a quartic residue modulo $p^k$ then $a$ is also a quartic residue modulo $p$. Conversely, suppose that $x^4 \equiv a \pmod p$. We will show by induction on $n$ that there is an integer $y_n$ such that $y_n^4 \equiv a \pmod{p^n}$.
This is true for $n = 1$ by the assumption that $a$ is a quartic residue modulo $p$.
Suppose that $y_n^4 \equiv a \pmod{p^n}$ for some $n$, and that $a$ is not divisible by $p$. Then $y_n^4 = k \cdot p^n + a$ for some integer $k$. We consider $$ (y_n + m \cdot p^n)^4 \equiv y_n^4 + 4 m p^n y_n^3 \equiv a + p^n (k + 4m y_n^3) \pmod{p^{n + 1}} $$ for various integers $m$. Here we note that $p^{2n}$, $p^{3n}$, and $p^{4n}$ are all divisible by $p^{n + 1}$ since $2n \geq n + 1 \iff n \geq 1$.
Since $a$ is not divisible by $p$, we also know that $y_n$ is not divisible by $p$, and so $4y_n^3$ is invertible modulo $p$. There is thus an integer $m$ such that $k + 4m y_n^3$ is divisible by $p$. Let $y_{n + 1} = y_n + m \cdot p^n$. Then we have that $y_{n + 1}^4 \equiv a \pmod{p^{n + 1}}$, and so $a$ is also a quartic residue modulo $p^{n + 1}$.
Of course there is nothing preventing you from investigating the function defined by $$ \left( \frac{a}{p} \right)_4^k. $$ This is indeed analogous to how the Jacobi symbol is defined, but you should then be aware that, just like with the Jacobi symbol, it's no longer true that $a$ is a [quadratic/quadratic] residue modulo $p^k$ if an only if the symbol evaluates to $1$. I don't offhand know of any resources that investigate this function.