Roots above the ground?

Applying Vieta-Cardano formulas: $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = (\alpha + \beta + \gamma + \delta)^2 - 2(\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta) = a^2 - 2a \Rightarrow$ The smallest value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is attained at the vertex $v$ of the parabola $a^2 - 2a = a(a - 2)$ which vertes is $v = 1$ ,i.e,

the smallest value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is the smallest value of $a^2 - 2a$ which is $\boxed{-1}$ attained at the vertex $v = 1$ of this parabola