Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does $p_{10}(x)$ have?
1. **Define the sequence of polynomials**:
   We start with \( p_0(x) = x \) and the recursive relation:
   \[
   p_{n+1}(x) = (p_n(x))^2 - \alpha
   \]

2. **Analyze the polynomial \( p_1(x) \)**:
   \[
   p_1(x) = (p_0(x))^2 - \alpha = x^2 - \alpha
   \]

3. **Analyze the polynomial \( p_2(x) \)**:
   \[
   p_2(x) = (p_1(x))^2 - \alpha = (x^2 - \alpha)^2 - \alpha
   \]

4. **General form of \( p_n(x) \)**:
   We observe that each polynomial \( p_n(x) \) is a composition of squares and subtractions of \(\alpha\). This recursive squaring and subtraction will generate polynomials of increasing degree.

5. **Degree of \( p_n(x) \)**:
   The degree of \( p_n(x) \) can be determined recursively:
   \[
   \deg(p_{n+1}(x)) = 2 \cdot \deg(p_n(x))
   \]
   Since \( \deg(p_0(x)) = 1 \), we have:
   \[
   \deg(p_n(x)) = 2^n
   \]

6. **Number of distinct real roots**:
   The polynomial \( p_n(x) \) has at most \( 2^n \) roots. However, we need to determine how many of these roots are distinct.

7. **Behavior of the sequence**:
   Given the recursive nature of the polynomials and the fact that \(\alpha\) is a root of the polynomial \( x^3 - 2x^2 + x - 1 \), we need to consider the fixed points of the sequence. Specifically, we need to check if \( p_n(x) = \alpha \) or \( p_n(x) = -\alpha \) for some \( n \).

8. **Fixed points and distinct roots**:
   Since \(\alpha\) is a unique real root of the polynomial \( x^3 - 2x^2 + x - 1 \), and given the recursive squaring, the sequence \( p_n(x) \) will generate polynomials that have roots converging to \(\alpha\) or \(-\alpha\). However, due to the nature of squaring, the roots will not be distinct.

9. **Conclusion**:
   The number of distinct real roots of \( p_{10}(x) \) is determined by the behavior of the sequence and the fixed points. Given the recursive squaring and the unique real root \(\alpha\), the number of distinct real roots of \( p_{10}(x) \) is 1.

The final answer is \(\boxed{1}\).