If $$\alpha, \beta \in C$$ are the distinct roots of the equation $$x^2 - x + 1 = 0$$, then $$\alpha^{101} + \beta^{107}$$ is equal to:

We start with the quadratic equation $$x^{2}-x+1=0$$ whose two distinct complex roots are denoted by $$\alpha$$ and $$\beta$$. Because the sum and product of the roots of the quadratic $$ax^{2}+bx+c=0$$ are given by the Vieta relations $$\alpha+\beta=-\dfrac{b}{a}$$ and $$\alpha\beta=\dfrac{c}{a}$$, we immediately have

$$\alpha+\beta=1 \quad\text{and}\quad \alpha\beta=1.$$

Placing either root back into the original equation gives a very useful reduction formula:

$$\alpha^{2}-\alpha+1=0\quad\Longrightarrow\quad\alpha^{2}=\alpha-1,$$

and, identically,

$$\beta^{2}=\beta-1.$$

Multiplying both sides of $$\alpha^{2}=\alpha-1$$ by $$\alpha$$ produces the cube of the root:

$$\alpha^{3}=\alpha(\alpha-1)=\alpha^{2}-\alpha=(\alpha-1)-\alpha=-1.$$

Thus

$$\alpha^{3}=-1\quad\Longrightarrow\quad\alpha^{6}=1.$$

Exactly the same reasoning provides $$\beta^{6}=1$$ as well.

Because $$\alpha^{6}=1$$, any power of $$\alpha$$ can be reduced modulo 6. We are asked to evaluate $$\alpha^{101}$$, so we divide the exponent by 6:

$$101 = 6\cdot16 + 5\quad\Longrightarrow\quad \alpha^{101} = \alpha^{6\cdot16+5} = (\alpha^{6})^{16}\,\alpha^{5}=1^{16}\,\alpha^{5}=\alpha^{5}.$$

To find $$\alpha^{5}$$ we rewrite it as $$\alpha^{3}\alpha^{2}$$ and substitute the values already obtained:

$$\alpha^{5}=\alpha^{3}\alpha^{2}=(-1)(\alpha-1)=-\alpha+1.$$

The same reduction is applied for $$\beta^{107}$$. First reduce the exponent:

$$107 = 6\cdot17 + 5\quad\Longrightarrow\quad \beta^{107} = \beta^{6\cdot17+5} = (\beta^{6})^{17}\,\beta^{5}=1^{17}\,\beta^{5}=\beta^{5}.$$

Again use $$\beta^{3}=-1$$ and $$\beta^{2}=\beta-1$$ to obtain

$$\beta^{5}=\beta^{3}\beta^{2}=(-1)(\beta-1)=-\beta+1.$$

Now we add the two required powers:

$$\alpha^{101}+\beta^{107}= (\,-\alpha+1\,)+(\,-\beta+1\,)= -(\alpha+\beta)+2.$$

Using the earlier Vieta result $$\alpha+\beta=1$$, we substitute:

$$-(\alpha+\beta)+2 = -1 + 2 = 1.$$

Hence, the correct answer is Option D.