----- > [!proposition] Proposition. ([[number of roots of a polynomial over integral domain cannot exceed degree]]) > Let $R$ be an [[integral domain]], and let $f \in R[x]$ be a [[polynomial 4|polynomial]] of degree $n$. Then the number of roots of $f$, counted with multiplicity, is at most $n$. ^proposition > [!proof]+ Proof. ([[number of roots of a polynomial over integral domain cannot exceed degree]]) The number of roots of $f$ in $R$ is less than or equal to the number of roots of $f$ viewed as a [[polynomial 4|polynomial]] over the [[field of fractions]] $K \supset R$, so it suffices to argue the result for $K$. > Now, $K[x]$ is a [[UFD]] (it's a [[field]]) and the roots of $f$ correspond to the [[irreducible element of an integral domain|irreducible]] factors of $f$ of degree $1$. Since the product of *all* irreducible factors of $f$ has degree $n$, the number of factors of degree $1$ cannot exceed $n$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```