Every non zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. ( Fundamental theorem of algebra, Wikipedia )
The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.
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| $x^2-1$ has $4$ roots... |
In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.
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