What is "the set of all polynomials in $\pi$"?

Question

From Ian Stewart's Galois Theory (2015, 4e, p. 20):

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

Answer 1

The terminology is hinted at in $(4)$.

A polynomial $p$ in $x$, with, say, integer coefficients, is defined by $$p(x)=\sum_{i=0}^{n}a_ix^i,$$ where $a_i\in\Bbb Z$ for all $i$, and for some $n\in\Bbb N\cup\{0\}$.

So let $x=\pi$ . . .

Answer 2

The set of all polynomials in $\pi$ with rational coefficients is the set of real numbers of the form $p(\pi)$, where $p(x)$ is a polynomial with rational coefficients; that is, it is the set $\{ p(\pi) \mid p(x) \in \mathbb{Q}[x] \}$.

Answer 3

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 $\pi$.

Answer 4

A polynomial in $\pi$ is in $4)$ an expression of the form $q_n\pi^n+q_{n-1}\pi^{n-1}+\dots +q_1\pi+q_0$, where each $q_i\in\mathbb Q$.

Or, as in $3)$, you could replace the $q_i\in\mathbb Q$ with $a_i\in\mathbb Z$, and have polynomials in $\pi$ with integer coefficients.

In these two cases we get a subring, but not a subfield, of $\mathbb C$.

On the other hand, we get a subfield when we consider all rational expressions in $\pi$; that is, quotients of polynomials in $\pi$ over $\mathbb Q$.