Theorem 1

The trace of an odd number of $\gamma$ matrices is zero.

Proof: For $n$ odd,

$$\textrm{Tr}[\not{\;\!\!a}_1\cdots\not{\;\!\!a}_n] = \textrm{Tr}[\not{\;\!\!a}_1\cdots\not{\;\!\!a}_n\gamma_5\gamma_5]... ...ma_5] = (-1)^n\textrm{Tr}[\not{\;\!\!a}_1\cdots\not{\;\!\!a}_n\gamma_5\gamma_5]$$

$$= 0 \quad\textrm{for $n$\ odd}.$$ (7.57)