In $\mathbb{Z}_6[X],$ factor each into two polynomials of degree 1: $g(X) = X + 2,$ $h(X) = X + 3$
$$g(X) = (3x+8)(2x+1)$$
$$h(X) = (6x+1)(x+3)$$
Does this contradict the unique factorization theorem? Why or why not?
I know that $\mathbb{Z}_6$ is not a field, and from our notes in class we defined the UFT to be: Each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F. Thus, can I conclude this does contradict the UFT since $g(X)$ and $h(X)$ can be factored into monic irreducible polynomials but $\mathbb{Z}_6$ is not a field? Any help in how to think about this question more clearly is helpful. Thank you!
It doesn't contradict the UFT because the theorem says "over a field" and, as you observe, $\mathbb{Z}_6$ is not a field. So "no prob."