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Are most prime numbers symmetric?

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2024-12-25 18:30:15

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If $n$ is a separable number, then it has property $X$ if the Galois group $\operatorname{Gal}(f_n(x)/\mathbb{Q})$ has property $X$ .

Hence we might say, a separable number is "abelian, cyclic, solvable, symmetric" if the corresponding Galois group is such.

I have computed with GP/PARI-SageMath for some primes the Galois groups, and it occurred to me, that with increasing primes $p$ the Galois groups get more and more the symmetric group. Since it is known that most random polynomials have symmetric Galois group, and it is also known (by Landaus interpretation of the Prime number theorem with the summatory Liouville function), that prime numbers behave like random values, one would expect, after seeing the empirical data, that most prime numbers have symmetric Galois group or are symmetric. This observation aligns with the empirical observation that primes behave randomly.

For primes $p>2$ of the form $f_p(x^2) = f_{f_p(4)}(x)$ I think I can prove that each of the prime divisors $q>2$ of $p-1$ is again of this form: $f_q(x^2) = f_{f_q(4)}(x)$ . Using Theorem 6.1 from here, the Galois group of those polynomials is transitive and imprimitive:

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