Why did they choose $x^4+1$ for the MixColumns step in Rijndael/AES?
The idea is that when you're multiplying two four-term (four-byte) polynomials together, you could wind up with a seven-byte polynomial, but you need to reduce it to another four-byte polynomial. They chose $x^4 + 1$ because

$x^i \mod (x^4 + 1) = x^{i \mod 4}$

See this for an example.