Discussion
Let \(\mathbf{F}_{q}\) be a finite field with q elements. Then which of the following need not be true ?
Answer: D
The statement that need not be true is that \(\mathbf{F}_{q}\) is algebraically closed; in fact it never is.
For any finite field, the polynomial \(\prod_{a \in \mathbf{F}_{q}} (x - a) + 1\) takes the value 1 at every element, so it has no root in \(\mathbf{F}_{q}\). A finite field therefore cannot be algebraically closed.
The other three are standard facts. The order of a finite field is a prime power \(p^{n}\). There is an irreducible polynomial of every degree n over \(\mathbf{F}_{q}\), because the extension \(\mathbf{F}_{q^{n}}\) exists and is simple. And the multiplicative group of any finite field is cyclic.
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