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So I have to give a irreducible factor decomposition of $11$ in $\Bbb Z[\sqrt3]$. I know that I have to use factors of the form $(a+b\sqrt{3})$ but I don't know how to do it :(

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    As this seems like a homework assignment; are there any theorems or examples about such a ring in your literature so far? – 2017-01-19

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HINT: You are looking for two factors of the form $a+b\sqrt{3}$ and $c+d\sqrt{3}$ such that

$$(a+b\sqrt{3})(c+d\sqrt{3})=11.$$

Note that $11=11+0\sqrt{3}$, thus you have to solve a system like this:

\begin{cases} ac+3bd=11\\ ad+cb=0 \end{cases}

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    If the system haven't solution means that $11$ is irreducible. I am sure that it isn't a square. – 2017-01-19
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    This is what I needed! I didn't have any examples. Thanks. – 2017-01-19
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    Ok, so this system does not have a solution where a,b,c,d are integers. It means 11 is irreducible in $Z[\sqrt{3}]$, right? – 2017-01-19
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    In this case there is a solution of the system – 2017-01-19