3

Suppose that a mathematician wrote,

Let $f_0, f_1, f_2, \cdots$ be functions in $\mathbb{Z}[X]$

What does $\mathbb{Z}[X]$ mean?

I am aware that $\mathbb{Z}$ is used to denote the set of integers.

In other words, $\mathbb{Z} = \{\cdots, -100000, \cdots, -3, -2, -1, 0, +1, +2, +3, \cdots, +100000, \cdots\}$

What on earth is $\mathbb{Z}[X]$?

Toothpick Anemone
  • 1,104
  • 4
    The set (ring) of polynomials with integer coefficients. – Randall Jul 14 '21 at 01:41
  • Polynomials with integer coefficients. Similarly, $\mathbb{Z} [i] ={\sum_{k=0}^{n} a_k i^k : n \in \mathbb{N} \ , a_k \in \mathbb{Z} } = {a+bi : a, b \in \mathbb{Z} } $. – Anonmath101 Jul 14 '21 at 01:41
  • 4
    Also, minor nitpick. An element $f \in \mathbb{Z}[X]$ isn't really a function per se, but a polynomial (which is a formal thing). The difference matters over other base rings, but it's not apparent over $\mathbb{Z}$. – Randall Jul 14 '21 at 01:48
  • 1
    A mathematician wouldn’t call them functions. – Thomas Andrews Jul 14 '21 at 02:22

1 Answers1

1

$\mathbb Z[X]$ is the ring of polynomials with integer coefficients. Written out in set-builder notation: $\mathbb Z[X]=\{\sum_{i=0}^{n} a_iX^i|i\in\mathbb N,a_i\in\mathbb Z \}$. You can plug in values of $\mathbb Z$ in for $X$ to get back an element of the base ring $\mathbb Z$. Though some authors may refer to a specific polynomial as "$f(x)\in \mathbb Z$," we consider polynomials as distinct algebraic objects here rather than functions – we are not considering the polynomial as a morphism/mapping between two groups/rings/fields in the same way we would think about a homomorphism. $X$ in this case is considered an indeterminate, not a variable (you can read more about this here).

$\mathbb Z[X]$ is a countably infinite ring (you can prove this by checking the ring axioms as an exercise!), but not a field – all polynomial rings are not closed under multiplicative inverses. This same notation applies to polynomial rings over other rings/fields. For example, $\mathbb Z_2[X]=\{\sum_{i=0}^{n} a_iX^i|i\in\mathbb N,a_i\in\mathbb Z_2 \}$ is the set of polynomials with coefficients $0$ or $1$. You can generate an ideal, a special subgroup of the polynomial ring, using one such $f(x)\in \mathbb Z$.

R.K.L
  • 86