I have been trying to solve the exercise 2.12 of the book "Algebraic number theory" published by Fröhlich and Cassels, but I fail to reach the conclusion. Let me sum up the statement
Let $K$ be a number field containing the $p$-roots of unity for $p$ a prime number. Let $\zeta$ be a primitive $p$-root of unity, and $\lambda=1-\zeta$. Fix $v$ a non-archimedean prime of $K$ dividing $p$. Consider $a$ an element of the form $a=1+\lambda^pc$ with $c\in\mathfrak o_v$ an integer for $v$. Then the extension $K(a^{\frac{1}{p}})/K$ is unramified at $v$, and for all $b\in K^*$ the norm residue symbol is given by the formula $$(a,b)_v = \zeta^{-S(\overline{c})v(b)}$$ where $\overline{c}$ is the $v$-residue of $c$ and $S$ is the trace from the residue field $k(v)$ to its prime field $\mathbb F_p$.
To do this, the hint goes as follow. Choose a $p$-root $\alpha$ of $a$ and write $\alpha = 1+\lambda x$. We may prove that $x$ is a root of a polynomial $f(X)\in \mathfrak o_v[X]$ with $f(X) = X^p-X-c$ mod $\mathfrak p_v$. With this, we deduce that the extension is indeed unramified. Moreover, writing $F$ the Frobenius morphism of this unramified extension, the fact that $f(X)$ kills $x$ allows us to show that $F(x) = x + S(\overline{c})$ mod $\mathfrak p_v$.
I was able to show all the content of the hint but I still fail to conclude. By definition, the norm residue symbol in the unramified case is given by $$(a,b)_v = \frac{F^{v(b)}(\alpha)}{\alpha}$$ But when I try computing this and using the hint, I fail to see how to make the term $\zeta$ appear with the suitable exponent. Would someone be able to give me a hand to reach the conclusion ?
Edit: to be more precise about my problem, the hint gives me information about the image of $x$ under the Frobenius mapping modulo $\mathfrak p_v$. However, if I decide to go modulo $\mathfrak p_v$ in the computation of the residue norm symbol, $\zeta$ being a $p$-root of unity it will be sent to $1$, and I will lose all relevant information for the computation. Because of this, I fail to see how I can make use of the hint.
The hint also makes mention of $\alpha' = \zeta\alpha$ another $p$-root of unity. If we write $\alpha' = 1 + \lambda x'$, then we obtain $x' = x - 1 $ mod $\mathfrak p_v$. In order to compute the norm residue symbol, we may use the $p$-root of $a$ that we want, be it $\alpha$ or $\alpha'$ or any other, but again I can't see how this would be relevant in the computation.