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Let $p>2$ be a prime number such that $p-1=2^st, s>0,t$ odd. Let $a,d\in \mathbb … {Z}^* /p \mathbb{Z}$ with $\left(\frac{a}{p}\right)=1$ and $\left(\frac{d}{p}\right)=-1$, where $\left(\frac{a}{p}\right)$ is a Legendre Symbol.

Prove that there exists even $k$ such that $k=log_{d^t}a^t$

My attempt:

I wrote the Legendre Symbol $\left(\frac{a}{p}\right)=1$ as $a^{\frac{p-1}{2}}=1$. This is the same as $d^{k(\frac{p-1}{2})}=1$ which is $d^{(p-1)k/2}=1^{k/2}$. This I believe solves the parity, but I don't know how to prove existence.

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