Exercise 1.3.14 (Evaluate $(ab, p^4)$ and $(a+b, p^4)$, if $(a, p^2) = p$ and $(b,p^3) = p^2$)

Question

Evaluate $(ab, p^4)$ and $(a+b, p^4)$ given that $(a, p^2) = p$ and $(b,p^3) = p^2$ where $p$ is a prime.

Richard Ganaye · Accepted Answer

\begin{proof} Given that $a \wedge p^2 = p$ and $ b\wedge p^3 = p^2$, we obtain 
$$a = p u, \ u \wedge p = 1,\qquad b = p^2 v,\ v \wedge p = 1.$$

Then $ab = p^3 w$, where $w = uv$ is prime to $p$. Therefore $$ab \wedge p^4 = p^3w \wedge p^4 = p^3(w \wedge p) = p^3,$$ so
$$ ab \wedge  p^4 = p^3.$$
Now $a +b = pu + p^2 v = p(u + pv) =ps$, where $s = u+pv$. Moreover $$s \wedge p = (u+pv) \wedge p = u \wedge p = 1.$$
Therefore $s \wedge p^3 = 1$. Then $(a+b) \wedge p^4 = ps \wedge p^4 = p(s \wedge p^3) = p$. So
$$(a+b) \wedge p^4 = p.$$
\end{proof}

Exercise 1.3.14 (Evaluate $(ab, p^4)$ and $(a+b, p^4)$, if $(a, p^2) = p$ and $(b,p^3) = p^2$)

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Exercise 1.3.14 (Evaluate $(ab, p^4)$ and $(a+b, p^4)$, if $(a, p^2) = p$ and $(b,p^3) = p^2$)

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