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$\text{lcm}$ of Polynomials in $(\mathbb{Z} / 2\mathbb{Z})[x]$

NUMTH-9TXZDN

Let $R$ be a commutative ring and $x, y \in R$. A common multiple of $x$ and $y$ is $m \in R$ such that both $x$ and $y$ divide m (i.e. there exist elements $a$ and $b$ of $R$ such that $ax = m$ and $by = m$), written $x | m$ and $y | m$, respectively. We say that $m$ is the least common multiple of $a$ and $b$, written $\text{lcm}(a, b)$, if it is minimal in the sense that for any other common multiple $n$ of $x$ and $y$, $m$ divides $n$.

Recall that $\mathbb{Z} / 2\mathbb{Z} = \{ \overline{0}, \overline{1} \}$ where $\overline{0} = \{ 2j~|~j \in \mathbb{Z}\}$ and $\overline{1} = \{2k + 1~|~k \in \mathbb{Z} \}$. Note that $\mathbb{Z} / 2\mathbb{Z}$ is a commutative ring and moreover that $\left(\mathbb{Z} / 2\mathbb{Z}\right)[x]$ (polynomials with coefficients in $\mathbb{Z} / 2\mathbb{Z}$) is a commutative ring.

Let $x + \overline{1}, x^2 + \overline{1} \in \left(\mathbb{Z} / 2\mathbb{Z}\right)[x]$ (where we utilize the convention $\overline{1}x = x$ and $\overline{1}x^2 = x^2$). What is $\text{lcm}\left(x + \overline{1}, x^2 + \overline{1}\right)$?

A

$x^3 + x^2 + x + \overline{1}$

B

$x^2$

C

$x + \overline{1}$

D

$x^2 + \overline{1}$