Proving $(a/p) = -1$ for infinitely many $p$, without Dirichlet's Theorem.

Question

I'm trying to find a proof of the following claim, without using Dirichlet's Theorem on the arithmetic progression of primes:

Claim: If $a$ is not a square, then there are infinitely many odd primes $p$ such that $\left(\frac ap\right) = -1$.

Note $(a/p)$ represents the Legendre symbol. Here's my current attempt:

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Proof: Clearly it suffices the prove the case where $a$ is squarefree. To begin with, let us prove the case where $a$ is odd.

Suppose we have already found such primes $p_1, \ldots, p_k$. Prime-factorise $a = r_1 \ldots r_s$ and note that $$\left( \frac a{p_i} \right) = -1 \implies (p_i, r_j) = 1 \ \ \forall i,j$$ Moreover, these are all odd primes, so all coprime to $8$. Thus by the Chinese Remainder Theorem, we can find $N > 1$ such that: $$ \begin{cases}N \equiv 1 \ (\text{mod} \ 8) \\ N \equiv 1 \ (\text{mod} \ p_i) \ \forall i \\ N \equiv 1 \ (\text{mod} \ r_j) \ \forall j < s \\ N \equiv c \ (\text{mod} \ r_s) \end{cases} $$

where $c$ is any quadratic non-residue modulo $r_s$.

Prime-factorise $N = q_1 \ldots q_l$. Then by construction these are all odd primes, and $q_i \ne p_j \ \forall i,j$.

Since $a$ is odd and $N \equiv 1 \ (\text{mod} \ 4)$, quadratic reciprocity gives:

$$ \prod_i \left(\frac a{q_i}\right) = \left(\frac{a}{N}\right) = \left(\frac{N}{a}\right) = \prod_{j=1}^{s-1} \left(\frac{N}{r_i}\right) \cdot \left(\frac{N}{r_s}\right) = \prod_{j=1}^{s-1} \left(\frac{1}{r_i}\right) \cdot \left(\frac{c}{r_s}\right) = -1 $$

So at least one of the terms in the product is $\left(\frac{a}{q_i}\right) = -1$, so $p_{k+1} = q_i$ is a new such prime, and the result is proven for $a$ odd.

Furthermore, we chose $N \equiv 1 \ (\text{mod} \ 8)$, so $$ \prod_j \left(\frac{2a}{q_j}\right) = \left(\frac{2}{N}\right)\left(\frac{a}{N}\right) = \left(\frac{a}{N}\right) = -1$$

and so by a similar argument the result follows for $a' = 2a$, hence for all even squarefree numbers.

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Is this proof valid? If so, can it be condensed or improved in any way?

Answer 1

A problem with the proof is that the construction of $N$ with CRT doesn't necessarily mean that it will also be squarefree (this was required when using quadratic residues in the proof). Also it's not true that $\left(\frac{a}{p_{1}}\right) \left(\frac{a}{p_{2}}\right) = \left(\frac{a}{p_{1}p_{2}}\right)$. A counter example being that $\left(\frac{-1}{3}\right) = -1$ and $\left(\frac{-1}{7}\right) = -1$ which means $\left(\frac{-1}{3}\right)\left(\frac{-1}{7}\right) = 1$. But $\left(\frac{-1}{21}\right) = -1$. Finally quadratic residue only applies when the top and the bottom of the legendre symbol are prime, so we can't have $\left(\frac{a}{N}\right) = \left(\frac{N}{a}\right)$.