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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?

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    $\begingroup$ It means that if you are given $x$ and $y$ elements of $F$, that there is exactly one element of $F$ which can be called $x+y$. $\endgroup$ – preferred_anon May 6 at 10:59
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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.

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  • $\begingroup$ I hope it's ok I inserted the word "necessarily" $\endgroup$ – J. W. Tanner May 6 at 17:44
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    $\begingroup$ @J.W.Tanner It's okay. $\endgroup$ – Javi May 7 at 11:39
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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...

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  • $\begingroup$ That's curious. Most definitions of binary operations I know always require the product to be uniquely defined? Or have I misunderstood you? $\endgroup$ – Allawonder May 6 at 19:42
  • $\begingroup$ @Allawonder: it's understood. It's implicit in the language. $\endgroup$ – TonyK May 6 at 20:36
  • $\begingroup$ How so, unless you've previously specified it? It's all language, isn't it, and don't we have to be sufficiently precise in order to communicate our meaning unambiguously? For example, are you saying it's impossible to define a binary operation that produces more than one element from two inputs? $\endgroup$ – Allawonder May 7 at 7:11
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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.

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  • $\begingroup$ I hope my (word change) edit was ok $\endgroup$ – J. W. Tanner May 6 at 17:42
  • $\begingroup$ @J.W.Tanner LOL! I don't even notice a difference; I still agree with it as it is, but I'm curious -- what change did you effect? $\endgroup$ – Allawonder May 6 at 19:43
  • $\begingroup$ If you click here, you can see I changed "definite" to "defined" $\endgroup$ – J. W. Tanner May 6 at 19:51
  • $\begingroup$ @J.W.Tanner Oh, yes. That's true. I think it's the same either way. To be definite is to be defined, specified, made unique, etc. I often find that using synonyms makes people realise that the hackneyed words have relatable meanings. Many people often seem to think the word well-defined is some magic word by mathematicians to make things happen by fiat any time they wish. $\endgroup$ – Allawonder May 7 at 7:15
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It means that for any $x, y, x', y'$, $x = x' \land y = y' \Rightarrow x + y = x' + y'$ and that $x = x' \land y = y' \Rightarrow xy = x'y'$.

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