Integral powers of the complexified universal real line

The base ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ is a path-connected paracompact Hausdorff CW complex and its tautological bundle is numerable. Under $P(\lambda )\cong {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$, the projective tautological line is $\lambda$, so the identity is a classifying map. Thus [F4] identifies $u=w_1(\lambda )$ with the nonzero polynomial generator. The real-linear map $({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}}\to \lambda \oplus \lambda$ given fiberwise by $v\otimes (a+ib)\mapsto (av,bv)$ has inverse $(x,y)\mapsto x\otimes 1+y\otimes i$ and commutes with the real transition functions, so it is a canonical real-bundle isomorphism by [F1]. Hence [F3] gives $w(({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}})=w(\lambda \oplus \lambda )=w(\lambda {)}^2=(1+u{)}^2=1+u^2$ over ${\mathbb{F}}_2$, where [F4] identifies $w(\lambda )=1+u$ and $2u=0$ in characteristic two.

Two-torsion: by [F5] with $j=0$ we have $2a=0$; multiplying by $a^{k-1}$ gives $2a^k=0$ for every $k\ge 1$.

The mod-two reduction of $a$: by [F2] applied to the complex bundle ${\lambda }_{\mathbb{C}}$, ${\rho }_2(a)=w_2(({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}})=w_2(\lambda \oplus \lambda )=u^2$ by step 1.1.

The class $a^k$ is nonzero for every $k\ge 1$: the cochain formula [F6] commutes with coefficient reduction, since reduction preserves products of values on each pair of faces. It therefore induces a ring homomorphism, so ${\rho }_2(a^k)={\rho }_2(a{)}^k=u^{2k}$ by step 2.1, and $u^{2k}\mathrm{\ne }0$ because $H^{\ast }({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)={\mathbb{F}}_2[u]$ is a polynomial ring by [F4].

Exact order two: by step 3.1 the element $a^k$ is nonzero, and by step 1.2 it satisfies $2a^k=0$, so its additive order is exactly two.

Boundary cases. For $k=1$ the statements read $2a=0$, ${\rho }_2(a)=u^2\mathrm{\ne }0$ and $\mathrm{ord}(a)=2$. The nonvanishing assertion concerns this universal line on the fixed nonempty base; step 1.1 identifies its class as a polynomial generator. The zeroth power is outside the assertion: $a^0=1$ has infinite integral order, so the restriction $k\ge 1$ is necessary. The coefficient field ${\mathbb{F}}_2$ is nonzero, so the nonzero reduction genuinely certifies nonvanishing over $\mathbb{Z}$. No orientation of $\lambda$ is used, since $w_1$ and the complexification are orientation-free. AC is used only through [A1].