I am reading about p-adic numbers, just begining on it. I came across this exercise:
Determine if the following equation has solutions in $\mathbb{Q}_5$. If this is the case, produce approximations $a_0 + a_1 5 + a_2 5^2 + a_3 5^3$ modulo $5^4$ in $\mathbb{Z}_5$ of both roots, with $a_i \in \{0, 1, 2, 3, 4, 5\}$
(a) $x^2 − 7 = 0$
(b) $x^2 + x − 7 = 0$
(c) $x^2 − 5^{−3} = 0$
Can someone show me what is the S.O.P (standard operating procedure) here? I t would help if you could do at least one as an example, I think there are two parts,
determine if solutions exists in $\mathbb{Q}_5$ and
If they do, find the approximations
