What is "the set of all polynomials in $\pi$"? From Ian Stewart's Galois Theory (2015, 4e, p. 20):

What does, for example, "the set of all polynomials in $\pi$" mean?

A "polynomial in $\pi$ with integer coefficient" is a slightly sloppy shorthand for "a number that is the value of some polynomial with integer coefficients, evaluated at $x=\pi$".
In other words, the set of all such numbers is the range of the evaluation morphism $\mathbb Z[X]\to\mathbb C$ that maps $X$ to $\pi$.
