IAT (IISER) 2020 | Complex Numbers Question 1 | Mathematics

Let $a, b$ be nonzero real numbers. If $p(x)=x^n+c_{n-1} x^{n-1}+\cdots+c_0$, where $c_{n-1}, \ldots, c_0$ are integers and $n \geq 3$, then which of the following statements is correct?

If $a+i b$ is a root of $p(x)$, then $n$ must be even.

If $a$ is a root of $p(x)$, then $n$ must be odd.

If $a+i b$ is a root of $p(x)$, then $(a+i b)^2$ can never be a root of $p(x)$.

If $a$ is a root of $p(x)$, then $a$ must be an integer.

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