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There is a simple criterion for detecting when $p(x)=x^4-7x^2-3x+1$ splits completely mod a prime $p$.

Let $$a_1=c_1=d_1=0,\ b_1=1$$ and set \begin{gather*} a_{n+1}=-d_n, \\ b_{n+1}=a_n+3d_n, \\ c_{n+1}=b_n+7d_n, \\ d_{n+1}=c_n, \\ A_n=\gcd(a_n-1,b_n,c_n,d_n), \end{gather*}

then the polynomial $p(x)$ splits completely mod $p$ if and only if $p\mid A_{p-1}$.

The recursion comes from the orbit of iterating the companion matrix of $p(x)$ and then taking a GCD. The method also works for algebraic integer of any degree which is not a root of unity.

There is a simple criterion for detecting when $p(x)=x^4-7x^2-3x+1$ splits completely mod a prime $p$.

Let $$a_1=c_1=d_1=0,\ b_1=1$$ and set \begin{gather*} a_{n+1}=-d_n, \\ b_{n+1}=a_n+3d_n, \\ c_{n+1}=b_n+7d_n, \\ d_{n+1}=c_n, \\ A_n=\gcd(a_n-1,b_n,c_n,d_n), \end{gather*}

then the polynomial $p(x)$ splits completely mod $p$ if and only if $p\mid A_{p-1}$.

The recursion comes from the orbit of iterating the companion matrix of $p(x)$ and then taking a GCD. The method also works for algebraic integer of any degree which is not a root of unity.

Added The method works even for degree one. If $p(x)=x-a, a \not= \pm 1$. Then $A_n=a^n-1$. Clearly $x-a$ splits completely mod every prime $p \nmid a$, so $p|A_{p-1}=a^{p-1}-1$. So the degree one reciprocity is just Fermat little theorem. The exact general definition of $A_n$ is also given in

https://math.stackexchange.com/questions/80265/splitting-of-primes-in-an-s-3-extension .

There is a simple criterion for detecting when $p(x)=x^4-7x^2-3x+1$ splits completely mod a prime $p$.

Let $$a_1=c_1=d_1=0,\ b_1=1$$ and set \begin{gather*} a_{n+1}=-d_n, \\ b_{n+1}=a_n+3d_n, \\ c_{n+1}=b_n+7d_n, \\ d_{n+1}=c_n, \\ A_n=\gcd(a_n-1,b_n,c_n,d_n), \end{gather*}

then the polynomial $p(x)$ splits completely mod $p$ if and only if $p\mid A_{p-1}$.

The recursion comes from the orbit of iterating the companion matrix of $p(x)$ and then taking a GCD. It is rather unsophisticated but it works for all algebraic integer of any degree which is not a root of unity.

There is a simple criterion for detecting when $p(x)=x^4-7x^2-3x+1$ splits completely mod a prime $p$.

Let $$a_1=c_1=d_1=0,\ b_1=1$$ and set \begin{gather*} a_{n+1}=-d_n, \\ b_{n+1}=a_n+3d_n, \\ c_{n+1}=b_n+7d_n, \\ d_{n+1}=c_n, \\ A_n=\gcd(a_n-1,b_n,c_n,d_n), \end{gather*}

then the polynomial $p(x)$ splits completely mod $p$ if and only if $p\mid A_{p-1}$.

The recursion comes from the orbit of iterating the companion matrix of $p(x)$ and then taking a GCD. The method also works for algebraic integer of any degree which is not a root of unity.

There is a simple criteria for detecting when $p(x)=x^4-7x^2-3x+1$ split completely mod a prime $p$.

Let $$a_1=c_1=d_1=0, b_1=1$$ and set $$a_{n+1}=-d_n,$$ $$b_{n+1}=a_n+3d_n,$$ $$c_{n+1}=b_n+7d_n,$$ $$d_{n+1}=c_n,$$ $$A_n=gcd(a_n-1,b_n,c_n,d_n),$$

then the polynomial $p(x)$ split completely mod $p$ if and only if $p|A_{p-1}$.

The recursion comes from the orbit of iterating the companion matrix of $p(x)$ and then taking a GCD. It is rather unsophisticated but it works for all algebraic integer of any degree which is not a root of unity.

There is a simple criterion for detecting when $p(x)=x^4-7x^2-3x+1$ splits completely mod a prime $p$.

Let $$a_1=c_1=d_1=0,\ b_1=1$$ and set \begin{gather*} a_{n+1}=-d_n, \\ b_{n+1}=a_n+3d_n, \\ c_{n+1}=b_n+7d_n, \\ d_{n+1}=c_n, \\ A_n=\gcd(a_n-1,b_n,c_n,d_n), \end{gather*}

then the polynomial $p(x)$ splits completely mod $p$ if and only if $p\mid A_{p-1}$.

The recursion comes from the orbit of iterating the companion matrix of $p(x)$ and then taking a GCD. It is rather unsophisticated but it works for all algebraic integer of any degree which is not a root of unity.