# Minimal polynomial

Given a field extension , if is algebraic over then the **minimal polynomial** of over is defined the monic polynomial of smallest degree such that . This polynomial is often denoted by , or simply by if is clear from context.

## Proof of existence/uniqueness

First note that as is algebraic over , there do exist polynomials with , and hence there must exist at least one such polynomial, say , of minimum degree. Now multiplying a polynomial by a scalar does not change it's roots, so we can find some nonzero such that is monic. Now by definition it follows that is *a* minimal polynomial for over . We now show that is is the only one.

Assume that there is some other monic polynomial such that and . By the division algorithm there must exist polynomials with such that . But now we have , which contradicts the minimality of unless . It now follows that . And now, as and are both monic polynomials of the same degree, it is easy to verify that , and hence . So indeed, is the *only* minimal polynomial for over .