More examples of fields

Two things today:

First, it was pointed out to me that I forgot to mention that in a field, 0\neq 1. This rules out having a field with exactly one element.

Second, here are some more examples of fields:

  • \mathbb Q, \mathbb R, \mathbb C as I mentioned last time.
  • Q[i]=\{a+bi\mid a,b\in\mathbb Q\}, the set of complex numbers with rational real and imaginary parts
  • \mathbb Z/p\mathbb Z, the integers 0, 1, … p-1 modulo p, where p is prime. Let’s look at these more carefully. The smallest example is when p=2:
The field of order 2 (called \mathbb Z/2\mathbb Z) has the following addition and multiplication tables:
\begin{tabular}{c|cc}+ & 0 & 1 \\\hline 0 & 0 & 1\\ 1 & 1 & 0\end{tabular}
\begin{tabular}{c|cc} \ensuremath{\times}&0 & 1 \\\hline 0 & 0 & 0\\ 1 & 0 & 1\end{tabular}
Here’s another field, this time with three elements:
\begin{tabular}{c|ccc}+ & 0 & 1 & 2 \\\hline 0 & 0 & 1 & 2\\ 1 & 1 & 2 & 0\\ 2& 2 & 0 & 1\end{tabular}
\begin{tabular}{c|ccc} \ensuremath{\times}&0 & 1 & 2\\\hline 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 2\\ 2 & 0 & 2 & 1\end{tabular}
You should take the time to verify that these rules for multiplication and addition do really make a field. It’s a worthwhile exercise, and you’ll feel more comfortable with how fields work.
For the curious, there are also fields with 4 and 5 elements! And 7, 8, and 9! What happened to 6 you ask? Well, there aren’t any. We’ll talk at some point about exactly what sizes finite fields can be. Currently we don’t have the tools to answer this question, and it isn’t really relevant on our path to Galois Theory.
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