There are two complete fields with respect to archimedean valuations – i.e. valuations for which |*x* + *y*| <= max{|*x*|, |*y*|} does not always hold – are **R** and **C**. This is because Ostrowski’s theorem says that every complete archimedean field containing **Q** must also contain **R**, i.e. be **R** or **C**, and if a field doesn’t contain **Q** then it must contain **F**(*p*), which admits no archimedean valuations.

Now, **C** is algebraically closed, so a polynomial over **C** is irreducible iff it is linear. **R** is a bit more complicated, but still easy because its algebraic closure is **C**, and [**C**:**R**] = 2. Suppose *f* is in **R**[*x*]. Then *f* splits into linear factors over **C**, and if *z* in **C** is a root of *f*, then so is * z*, the complex conjugate of

*z*; this is because complex conjugation is a field isomorphism from

**C**to itself that fixes every element of

**R**, so that it preserves polynomial equations with coefficients in

**R**.

Now, (*x* – *z*)(*x* – * z*) =

*x*^2 – (

*z*+

*) +*

__z__*z*, where

__z__*z*+

*and*

__z__*z*are always real, so if

__z__*f*has a proper complex root, then it has a factor of degree 2. If it has no proper complex roots, then it of course splits into linear factors. In other words, every irreducible polynomial over

**R**has degree 1 or 2. Further, a linear polynomial is always irreducible, while a quadratic

*ax*^2 +

*bx*+

*c*is irreducible iff its two complex roots are proper complex, i.e. iff (-

*b*(+/-) SQRT(

*b*^2 – 4

*ac*))/2

*a*is not real, i.e. iff

*b*^2 < 4

*ac*.

Local fields are a lot more involved, because their algebraic closures have infinite degree over them. The same of course applies to global fields – recall that **A** has infinite degree over **Q**, for instance. Hensel’s lemma provides one local field analog of polynomial factorization in **R**, which more or less corresponds to the statement that every real polynomial of odd degree has a real root.

There’s another lemma that vaguely resembles the *b*^2 < 4*ac* condition, which is also useful in constructing extensions of local fields (more on this later): if K is a local field and *f* is an irreducible polynomial over K, *a*(*n*)*x*^*n* + … + *a*0, then |*a*(*i*)| <= max{|*a*0|, |*a*(*n*)|} for all *i*. Equivalently, the coefficient of *f* whose valuation is the highest is either *a*0 or *a*(*n*).

To see why, first multiply by a suitable power of a prime element *p* in the discrete valuation ring R of K, so that *f* becomes a polynomial over R. If all coefficients of *f* are in P, then divide by some power of *p* to make *f* have at least one coefficient not in P. In this configuration, the coefficients of *f* whose valuation is the highest must have valuation exactly 1, so that they’re the only ones outside P.

Now, let *j* be the highest index for which |*a*(*j*)| = 1. It suffices to show that *j* = 0 or *n*. But if 1 <= *j* <= *n*-1, then we can reduce *f* mod P; as *a*(*i*) is in P for *i* > *j*, the lowest term in *f* with nonzero coefficient mod P is *x*^*j*. But then *f* = *x*^*j*(*a*(*n*)*x*^(*n*–*j*) + … + *a*(*j*)) mod P, and *x*^*j* and *a*(*n*)*x*^(*n*–*j*) + … + *a*(*j*) are coprime since *a*(*j*) != 0 mod P. Also, *j* and *n*–*j* are not zero, since by assumption 1 <= *n* <= *n*-1. Therefore, by Hensel’s lemma, *f* is reducible as the product of polynomials equivalent to *x*^*j* and *a*(*n*)*x*^(*n*–*j*) + … + *a*(*j*) mod P, contradicting the assumption that *f* is irreducible.

Note that this lemma says |*a*(*i*)| <= max{|*a*0|, |*a*(*n*)|}, not |*a*(*i*)| < max{|*a*0|, |*a*(*n*)|}. That is, it’s entirely possible for a coefficient other than *a*0 and *a*(*n*) to attain the maximum valuation, as long as it shares it with *a*0 or *a*(*n*).