any linear non-constant polynomial in $F[x]$ is irreducible?

Question

Let $F$ be a field.

Use the formula $\deg(fg) = \deg (f) + \deg (g)$ to show that any linear non-constant polynomial in $F[x]$ is irreducible.

Thanks!

Answer 1

If $L$ is a linear polynomial, then $\text{deg }L = 1$.

If $L=fg$, then $\text{deg }L =\text{deg }f + \text{deg }g$, which is to say that $1 = \text{deg }f +\text{deg }g$, where $\text{deg }f, \text{deg }g \geq 0$.

Now, what are the possible combinations of $\text{deg }f$ and $\text{deg }g$?

Can you conclude, given that constants are polynomials of degree zero?