TITLE: Zero polynomial
QUESTION [6 upvotes]: Possible Duplicate:
Polynomial of degree $-\infty$? 

Today in Abstract Algebra my instructor briefly mentioned that sometimes the zero polynomial is defined to have degree $-\infty$. What contexts have caused this to become convention?

REPLY [3 votes]: Here’s a viewpoint that puts the question in much broader context. In a discretely valued field, like $\mathbb{Q}$ with the $p$-valuation $v_p$ that counts the divisibility by the prime $p$, defined so that $v_p(p)=1$ and $v_p(m)=0$ for integers prime to $p$, and additive in the sense that $v_p(\lambda\mu)=v_p(\lambda)+v_p(\mu)$, it’s universal to extend $v$ so that $v(0)=+\infty$. Then the relation $v(\lambda+\mu)\ge\min(v(\lambda),v(\mu))$ holds throughout the field.
Well, just as $\mathbb{Q}$ has many discrete valuations, so does the field $k(X)$ of rational functions over a field $k$. One nice valuation is defined by setting, for a polynomial $P(X)$, $v(P)=-\deg(P)$, and extending by additivity, so that that $v(P/Q)=\deg(Q)-\deg(P)$, both $P$ and $Q$ being polynomials. This $v$ measures the zeroness of a rational function at $\infty$. And in accordance with the standard practice for valued fields, $v(0)=+\infty$, fitting in perfectly with $\deg(0)=-\infty$.