= Bott element
{c}
{title2=$\beta$}
Let $H$ be the <tautological bundle> on $\mathbb{CP}^1\cong S^2$. One sign convention for the Bott element is
$$
\beta=[H]-[1]\in\widetilde K^0(S^2),
$$
where $[1]$ is the rank-one <trivial vector bundle>. It has rank zero, and its <Chern character> has degree-two term $c_1(H)$, a generator up to sign. Choosing the dual line bundle reverses that sign.
The $n$-fold reduced <external product in topological K-theory> gives a generator of
$$
\widetilde K^0((S^2)^{\wedge n})\cong\widetilde K^0(S^{2n})\cong\mathbb Z.
$$
This product uses the <smash product> of different sphere factors. It is distinct from taking <exterior powers> of a <vector bundle> over one fixed base. Multiplication by the Bott element realizes <Bott periodicity>.<Vector Bundles and K-Theory by Allen Hatcher>[Corollary 2.12, page 55]{ref}
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