Discussion
Which of the following is an integral domain but not a field ?
Answer: C
The integral domain that is not a field is \(\mathbf{R}[x]\).
The polynomial ring over a field has no zero divisors, since the leading terms of two non-zero polynomials multiply to a non-zero term, so it is an integral domain. But \(x\) has no inverse: \(x \cdot p(x)\) always has degree at least 1, never equal to 1. So it is not a field.
\(\mathbf{Z}_{10}\) is not even an integral domain (\(2 \cdot 5 = 0\)). \(\mathbf{Z}_{31}\) is a field because 31 is prime, and \(\mathbf{C}(x)\), the field of rational functions, is a field by construction.
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