Algebraic Number Theory and Fermat's Last Theorem, 4th Edition
Lemma 1.7. (Gauss’s Lemma.) Let \(p\in\mathbb Z[t]\) and suppose that \(p=gh\) where \(g,h\in\mathbb Q[t]\). Then there exists \(\lambda\in\mathbb Q,\lambda\ne0\), such that \(\lambda g,\lambda^{-1}h\in\mathbb Z[t]\).
Corollary 1.6. An irreducible polynomial over a subfield \(K\) of \(\mathbb C\) has no repeated zeros in \(\mathbb C\).
Proof. Suppose \(f\) has repeated roots. Then by Theorem 1.5, \(f\) and \(\text Df\) have a common factor of degree \(>0\). But this contradicts with the fact that \(f\) is irreducible. \(\square\)
Theorem 1.5. Let \(K\) be a field of characteristic zero. A non-zero polynomial \(f\) over \(K\) is divisible by the square of a polynomial of degree \(>0\) if and only if \(f\) and \(\text Df\) have a common factor of degree \(>0\).
Proof. (\(\Rightarrow\)) Let \(g\) be a polynomial of degree \(>0\) such that \(f=g^2h\) for some polynomial \(h\). Then \(\text Df=2gh+g^2\text Dh\) and hence \(f\) and \(\text Df\) have a common factor \(g\) of degree \(>0\).
(\(\Leftarrow\)) Suppose there is no factor of \(f\) that is a square, so \(f=f_1f_2\cdots f_n\) where \(f_i\)s are polynomials that are irreducible and \(\gcd(f_i,f_j)=1\) for \(i\ne j\). Then \(\text Df=\sum_{1\le i\le n}(\text Df_i\prod_{j\ne i}f_j)\). Since \(f_i\) is irreducible, \(\gcd(f_i,\text Df_i)=1\), so \(\gcd(f,\text Df)=1\). \(\square\)
