Examples of Applying Hensel's Lemma
Recall from the Hensel's Lemma page that if $f$ is a polynomial, $p$ is a prime, $x = a$ is a solution to $f(x) \equiv 0 \pmod {p^k}$, and $f'(a) \not \equiv 0 \pmod p$ then there exists a unique lift to a solution $a + tp^k$ where $f(a + tp^k) \equiv 0 \pmod {p^{k+1}}$. In particular, if $x = a_1$ is a solution to $f(x) \equiv 0 \pmod p$ and $f'(a_1) \not \equiv 0 \pmod p$ then the recursive formula:
(1)is a solution to $f(x) \equiv 0 \pmod {p^{k+1}}$. We will now look at an example of applying Hensel's Lemma to solving $f(x) \equiv 0 \pmod {p^k}$.
Example 1
Use Hensel's Lemma to find a solution to $x^3 - 2x \equiv 1 \pmod {125}$.
Let $f(x) = x^3 - 2x - 1$. We first find a solution to $f(x) \equiv 0 \pmod 5$. We see that:
(2)So $a_1 = 2, 3, 4$ are all solutions to $f(x) \equiv 0 \pmod 5$. We compute the derivative of $f$:
(3)Observe that:
(4)We cannot apply Hensel's Lemma to $a_1 = 2, 3$, but we can apply Hensel's Lemma to $a_1 = 4$. Observe that $[f'(4)]^{-1} = 1 \pmod 5$. So:
(5)So $x = 124$ is a solution to $x^3 - 2x \equiv 1 \pmod {125}$.
Example 2
Use Hensel's Lemma to find all solutions to $x^4 + x^3 + 2x^2 + x \equiv 13 \pmod {343}$.
Let $f(x) = x^4 + x^3 + 2x^2 + x - 13$. Then $f(x) \equiv 0 \pmod {7^3}$. We attempt to find a solution to $f(x) \equiv 0 \pmod 7$. We have that:
(7)Let $s = 2$ and $t = 4$. Then $s$ and $t$ are solutions to $f(x) \equiv 0 \pmod p$. Now the derivative of $f(x)$ is:
(8)Plugging in $s$ and $t$ into $f'$ gives us:
(9)So Hensel's Lemma can be applied to both $s$ and $t$.
First set $a_1 = 2$. From above we have that $f'(2) = 4$. So $[f'(2)]^{-1} = [4]^{-1} = 2 \pmod 7$. Hence:
(10)And:
(11)So $x = 205$ is a solution to $f(x) \equiv 0 \pmod {7^3}$.
Setting $a_1 = 4$ and using the same process shows that $x = 4$ is also a solution to $f(x) \equiv 0 \pmod {7^3}$.