note-for-irreducible-polynomials

No, $F[X]$ is not the notation for a field itself. Instead, $F[X]$ is the notation for the polynomial ring over a field $F$.

Given a field $F$, $F[X]$ represents the set of all polynomials with coefficients in $F$. The variable $X$ is an indeterminate, which means it doesn’t take on any specific value within the field $F$.

For instance, if $F$ is the field of real numbers (often denoted as $\mathbb{R}$), then $F[X]$ or $\mathbb{R}[X]$ represents the set of all polynomials with real coefficients.

Although $F[X]$ is a ring, it’s not generally a field because most polynomials don’t have multiplicative inverses in $F[X]$. For example, in $\mathbb{R}[X]$, the polynomial $X$ doesn’t have a multiplicative inverse, because there’s no polynomial you can multiply $X$ by to get 1 (the multiplicative identity in $\mathbb{R}[X]$). However, $F(X)$ (note the parentheses) denotes the field of rational functions over $F$, and this is a field.