In the definition of a field, what means unique element $x+y$ and $xy$?

Question

In my linear algebra textbook, field is defined like below:

A field F is a set on which two operations $+$ and $\cdot$ (addition and multiplication, respectively) are defined so that, for each pair of elements $x,y$ in $F$, there are unique elements $x+y$ and $x\cdot y$ in $F$ for which following conditions hold for all elements $a,b,c$ in F

(the rest omitted)

What does it mean unique elements in here? It means $x+y$ and $x\cdot y$ is distinct?

Answer 1

It means that $x+y$ represents only one element and $x\cdot y$ represents only one element too. That doesn't necessarily mean that the unique element $x+y$ is a different element from the unique element $x\cdot y$.

For example, $2+2$ is a unique number and so is $2\cdot 2$, but $2+2=2\cdot 2=4$.

To show an example when an operation might yield non unique results, consider the square root of a positive real number. For instance, $\sqrt{2}$ in general means the positive root, but if we define the square root of $2$ as a number $x$ satisfying $x^2=2$, then there are 2 different solutions.

Answer 2

The word unique means exactly one of some object; in other words, in a field we require the binary operations to always yield a single value. Then we say the operations are properly defined.

Answer 3

It means that for any $x, y, x', y'$, $x = x' \land y = y' \Rightarrow x + y = x' + y'$ and that $x = x' \land y = y' \Rightarrow xy = x'y'$.

Answer 4

The word unique means absolutely nothing here, and can only cause confusion (as your question attests). It should say:

for each pair of elements $x,y$ in $F$, there are elements $x+y$ and $x\cdot y$ in $F$ for which...