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Which prime numbers in $\mathbb Z$ are precisely the prime elements of $\mathbb Z[i]$ and why?

In my book it has been left as an exercise to show that a prime number in $\mathbb Z$ is a prime element of $\mathbb Z[i]$ if and only if it is of the form $4k + 3$.

How can I proceed?Please help me.

Thank you in advance.

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    What are your thoughts on the question? What have you tried? Try looking at an example prime of $\Bbb Z$ that fails to be prime in $\Bbb Z[i]$, can you see how it should be factored? – 2017-01-12
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    See answer $(1)$ of http://math.stackexchange.com/questions/458795/whats-are-all-the-prime-elements-in-gaussian-integers-mathbbzi – 2017-01-12
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    One solution is presented [here](http://www2.math.ou.edu/~kmartin/nti/chap6.pdf) – 2017-01-12
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    The prime elements which are of the form $4k + 1$ can be expressed as the sum of two squares due to Fermat's theorem.Though I don't understand the proof clearly.In fact I have understood two proposition of Euler from wikipedia.But I have stuck when I tried to understand the lemma which asserts that "If $a$ and $b$ are relatively prime to each other then each factor of $a^2 + b^2$ can be expressed as the sum of two squares." Can you please help me understanding it from wikipedia?Then it will really help me a lot. – 2017-01-12

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