# Why do odd dimensions and even dimensions behave differently?

It is well known that odd and even dimensions work differently.

Is there some unifying reason for these differences? Why is dimensional parity so important?

For real vector spaces, i.e. $\mathbb{R}^N$, one important difference between odd and even $N$ is that every real polynomial with odd order has a real zero, while there are real polynomials with even order that only have complex zeros.
Thus, a real invertible $N\times N$ matrix always has a non-zero eigenvector if $N$ is odd (because the characteristic polynomial has order $N$, and by the above has a zero). For even $N$, however, there are invertible real matrices with no non-zero eigenvector.