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Does a finite field need a multiplicative identity?

In a finite field $\mathbb{F}$ of characteristic $p$ and size $q$, do we need $\mathbb{F}$ to have an element $0$ such that $0\cdot a = 0$ for all $a$?
I believe that this can be done, by defining $x=\prod_{i=1}^m a_i^{q_i}$ and considering the action of the automorphism $\alpha$ which is the map that fixes the elements of $\mathbb{F}$ and acts as $x\mapsto x^{q-1}$. But how to show that this is an automorphism? Can we just construct a multiplicative inverse for $\alpha$ and use the multiplicative inverse to define the zero?

A:

Yes, the requirement that the finite field have a zero element is equivalent to requiring that the field have a multiplicative group of order $q$. The requirement that the group be cyclic is equivalent to requiring that the multiplicative group be abelian, and in fact in that case the group is isomorphic to the additive group of $\Bbb F_q$, and the corresponding field is the quotient field of $\Bbb F_q$.
Your description is not quite right, however, because the multiplicative group of $\Bbb F_q$ is cyclic, but $\Bbb F_q$ itself is not cyclic. Also, an automorphism is an element $f$ of the eea19f52d2

Developed by one proactive French security research that goes by the name of Adrien Guinet, Wannakey is a tiny tool that should be able to recover the RSA private keys used by WannaCry on your computer. We’re pointing out once again that it only works if you haven’t rebooted your computer after the infection.
Requires you to follow a fairly straightforward set of procedures and its success rate might have to do with luck more than with anything else
Of course, it goes without saying that it can’t be expected of this tool to have an outstanding success rate. Hence, pray that you’re a lucky person and hope that the associated memory has not been moved or deleted.
Right, here’s what you need to do: first off, make sure that Microsoft Visual C++ Redistributable Package 2013 is present on your computer’s system. Secondly, you need to locate the process identifier of the wrcy.exe (WannaCry) process using the Task manager, and locate a certain 00000000.pky file.
Now, download the utility from our servers and launch it using your computer’s CMD by using your own custom expression based on this example: “search_primes.exe PID pathto00000000.pky”. The priv.key file will be generated in the current directly, of course, if a valid prime number is found in your computer’s memory.
Fingers crossed and good luck!
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Wannakey Description:
Developed by one proactive French security research that goes by the name of Adrien Guinet, Wannakey is a tiny tool that should be able to recover the RSA private keys used by WannaCry on your computer. We’re pointing out once again that it only works if you haven’t rebooted your computer after the infection.
Requires you to follow a fairly straightforward set of procedures and its success rate might have to do with luck more than with anything else
Of course, it goes without saying that it can’t be expected of this tool to have an outstanding success rate. Hence, pray that you’re a lucky person and hope that the associated memory has not been moved or deleted.
Right, here’s what you need to do: first off, make sure that Microsoft Visual C++ Redistributable Package 2013 is present on your computer’s system. Secondly, you need to locate the process identifier of