Let $P(a, b)$, be a polynomial with monomials all degree $1$ or higher and integer coefficients (in other words, there are no constant terms). Prove that if $P(a, b) = 1$ (when replacing $a$ and $b$ with integers), $a$ and $b$ are relatively prime. Thanks for help on proving this.
The converse of this (is easy to prove), If $a$ and $b$ are relatively prime, then there exists a function $f(a, b) = P(a, b)$ (where $P(a, b)$ is a polynomial) such that $f(a, b) = 1$. $P(a, b) = ax - by = 1$ where $x$ and $y$ are coefficients. (This comes from Bézout's identity).
Assume $a$ and $b$ are not relatively prime, i.e., $\gcd(a,b)=d>1.$ Then $d$ must divide $P(a,b),$ since every term in the polynomial is either a multiple of $a$ or of $b.$ Thus, $P(a,b)\neq 1.$
Note that we can write $P(a,b)=a P_1(a,b)+b P_2(a,b)$ for integer polynomials $P_1$ and $P_2.$