Ideals, Quotient Rings, and Fields
Let's take a formal approach to finite fields using ideals and quotient rings.
In our quest to determine the cycle length and delay of our cellular automaton, we figured out that each iteration corresponded to a matrix multiplication. To study the game, we looked at the powers of this matrix. Our first attempt with diagonalization solved the problem 80% of the time, but diagonalization seems to have failed when the number of cells in a row was a multiple of five. We decided that we should consider powers of the matrix directly, which led us to the Cayley-Hamilton Theorem.
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The Cayley-Hamilton Theorem states that every square matrix is a solution to its characteristic equation. We just proved the Cayley-Hamilton Theorem for the complex numbers and its subrings in the previous article, but not for finite fields. Furthermore, our proof relied pretty heavily on the topology of complex numbers, which feels like cheating. To fix both these issues, we need to come up with a consistent way to talk about rings other than subrings of complex numbers. These rings are known as quotient rings.
Quotient Groups
We’ve already discussed quotient groups in a previous article, but not to the extent…
