Discussion
Let F be a field with at least two elements. Then which of the following statements need not be true ?
Answer: D
The statement that need not be true is If F is infinite, then \(\mathbf{Z}_{p}\) cannot be embedded in F for any prime \(p\).
The field of rational functions \(\mathbf{Z}_{p}(x)\), or the algebraic closure of \(\mathbf{Z}_{p}\), is infinite but has characteristic \(p\) and contains \(\mathbf{Z}_{p}\).
The other three always hold. Every field contains its prime subring, which is \(\mathbf{Z}\) (characteristic 0) or \(\mathbf{Z}_{p}\) (characteristic \(p\)), giving A. A field has no zero divisors, and \(\mathbf{Z}_{n}\) for composite \(n\) has them, so it cannot sit inside F (B). And if \(\mathbf{Z}\) embeds, every nonzero integer is invertible in F, so all fractions \(a/b\) are there and Q embeds (C).
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