? is an algebraically closed field

Do not be intimidated by the terminology: a field is simply a set in which addition, subtraction, multiplication and division can be performed according to the usual rules.

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$\mathbb{C}\backslash mathbb\{C\}$, analytically complete, algebraically incomplete --------------------------------------------------------------------------

Let's begin with the set $\mathbb{Q}\backslash mathbb\{Q\}$ of rational numbers (quotients of two integers), which represents a kind of completion not shared by $\mathbb{N}\backslash mathbb\{N\}$ or $\mathbb{Z}\backslash mathbb\{Z\}$, the sets of positive integers and integers, respectively. The four operations have their usual properties there (associativity, commutativity and distributivity), and their results also remain in $\mathbb{Q}\backslash mathbb\{Q\}$. In other words, this set is closed under these operations, except that division by zero is impossible. In short, $\mathbb{Q}\backslash mathbb\{Q\}$ is a field and is “complete” in this sense.

The field $\mathbb{Q}\backslash mathbb\{Q\}$ is suitable for practical purposes when approximations suffice. However, it contains no number corresponding to certain simple geometric quantities, such as the length of the diagonal of a unit square. That length can only be approximated. We can obtain approximations that come as close to the value as desired, for example 1.5, 1.4, 1.414…, but no rational number whose square is exactly 2. In other words, 2 has no square root in $\mathbb{Q}\backslash mathbb\{Q\}$.