Step 1: Use the sum and product of roots:
$\alpha+\beta=-1$ and $\alpha\beta=1$. Because $\alpha\beta=1$, we get $\beta=1/\alpha$, so $\alpha^n+\beta^n=\alpha^n+\alpha^{-n}$.
Step 2: Use the order of alpha:
From $\alpha^2+\alpha+1=0$, multiplying by $\alpha-1$ gives $\alpha^3=1$. The same holds for $\beta$, so $\beta^3=1$. Since $2026=3(675)+1$, we get $\alpha^{2026}=\alpha$ and $\beta^{2026}=\beta$.
Step 3: Add:
$\alpha^{2026}+\beta^{2026}=\alpha+\beta=-1$, option A.
Final Answer:
Powers repeat every 3, so the sum equals alpha plus beta, which is -1.
\[ \boxed{\text{(A) }-1} \]