Eisenstein's criterion

Template:Short description In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients.

This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial.

This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it.

Criterion

Suppose we have the following polynomial with integer coefficients: $$Q(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0.$$

If there exists a prime number Template:Mvar such that the following three conditions all apply:

• Template:Mvar divides each a_i for 0 ≤ i < n,
• Template:Mvar does not divide a_n, and
• p² does not divide a₀,

then Template:Mvar is irreducible over the rational numbers. It will also be irreducible over the integers, unless all its coefficients have a nontrivial factor in common (in which case Template:Mvar as integer polynomial will have some prime number, necessarily distinct from Template:Mvar, as an irreducible factor). The latter possibility can be avoided by first making Template:Mvar primitive, by dividing it by the greatest common divisor of its coefficients (the content of Template:Mvar). This division does not change whether Template:Mvar is reducible or not over the rational numbers (see Primitive part–content factorization for details), and will not invalidate the hypotheses of the criterion for Template:Mvar (on the contrary it could make the criterion hold for some prime, even if it did not before the division).

Examples

Eisenstein's criterion may apply either directly (i.e., using the original polynomial) or after transformation of the original polynomial.

Direct (without transformation)

Consider the polynomial Q(x) = 3x⁴ + 15x² + 10. In order for Eisenstein's criterion to apply for a prime number Template:Mvar it must divide both non-leading coefficients 15 and 10, which means only p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10. One may therefore conclude that Template:Mvar is irreducible over Q (and, since it is primitive, over Z as well). Note that since Template:Mvar is of degree 4, this conclusion could not have been established by only checking that Template:Mvar has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible.

Indirect (after transformation)

Often Eisenstein's criterion does not apply for any prime number. It may however be that it applies (for some prime number) to the polynomial obtained after substitution (for some integer Template:Mvar) of x + a for Template:Mvar. The fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift.

For example, consider H = x² + x + 2, in which the coefficient 1 of Template:Mvar is not divisible by any prime, Eisenstein's criterion does not apply to Template:Mvar. But if one substitutes Template:Mvar for x + 3 in Template:Mvar, one obtains the polynomial x² + 7x + 14, which satisfies Eisenstein's criterion for the prime number 7. Since the substitution is an automorphism of the ring Q[x], the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally. In this particular example it would have been simpler to argue that Template:Mvar (being monic of degree 2) could only be reducible if it had an integer root, which it obviously does not; however the general principle of trying substitutions in order to make Eisenstein's criterion apply is a useful way to broaden its scope.

Another possibility to transform a polynomial so as to satisfy the criterion, which may be combined with applying a shift, is reversing the order of its coefficients, provided its constant term is nonzero (without which it would be divisible by Template:Mvar anyway). This is so because such polynomials are reducible in R[x] if and only if they are reducible in R[x, x⁻¹] (for any integral domain R), and in that ring the substitution of x⁻¹ for Template:Mvar reverses the order of the coefficients (in a manner symmetric about the constant coefficient, but a following shift in the exponent amounts to multiplication by a unit). As an example 2x⁵ − 4x² − 3 satisfies the criterion for p = 2 after reversing its coefficients, and (being primitive) is therefore irreducible in Z[x].

Cyclotomic polynomials

An important class of polynomials whose irreducibility can be established using Eisenstein's criterion is that of the cyclotomic polynomials for prime numbers Template:Mvar. Such a polynomial is obtained by dividing the polynomial x^p − 1 by the linear factor x − 1, corresponding to its obvious root 1 (which is its only rational root if p > 2): $$\frac{x^p-1}{x-1}=x^{p-1}+x^{p-2}+\cdots +x+1.$$

Here, as in the earlier example of Template:Mvar, the coefficients 1 prevent Eisenstein's criterion from applying directly. However the polynomial will satisfy the criterion for Template:Mvar after substitution of x + 1 for Template:Mvar: this gives $$\frac{(x+1{)}^p-1}{x}=x^{p-1}+(\genfrac{}{}{0ex}{}{p}{p-1})x^{p-2}+\cdots +(\genfrac{}{}{0ex}{}{p}{2})x+(\genfrac{}{}{0ex}{}{p}{1}),$$ all of whose non-leading coefficients are divisible by Template:Mvar by properties of binomial coefficients, and whose constant coefficient is equal to Template:Mvar, and therefore not divisible by p². An alternative way to arrive at this conclusion is to use the identity (a + b)^p = a^p + b^p which is valid in characteristic Template:Mvar (and which is based on the same properties of binomial coefficients, and gives rise to the Frobenius endomorphism), to compute the reduction modulo Template:Mvar of the quotient of polynomials: $$\frac{(x+1{)}^p-1}{x}\equiv \frac{x^p+1^p-1}{x}=\frac{x^p}{x}=x^{p-1}(\mathrm{mod}p),$$ which means that the non-leading coefficients of the quotient are all divisible by Template:Mvar; the remaining verification that the constant term of the quotient is Template:Mvar can be done by substituting 1 (instead of x + 1) for Template:Mvar into the expanded form x^p−1 + ... + x + 1.

History

Theodor Schönemann was the first to publish a version of the criterion, in 1846 in Crelle's Journal, which reads in translation

That (x − a)ⁿ + pF(x) will be irreducible to the modulus p² when F(x) to the modulus p does not contain a factor x−a.

This formulation already incorporates a shift to Template:Mvar in place of 0; the condition on F(x) means that F(a) is not divisible by Template:Mvar, and so pF(a) is divisible by Template:Mvar but not by p². As stated it is not entirely correct in that it makes no assumptions on the degree of the polynomial F(x), so that the polynomial considered need not be of the degree Template:Mvar that its expression suggests; the example x² + p(x³ + 1) ≡ (x² + p)(px + 1) mod p², shows the conclusion is not valid without such hypothesis. Assuming that the degree of F(x) does not exceed Template:Mvar, the criterion is correct however, and somewhat stronger than the formulation given above, since if (x − a)ⁿ + pF(x) is irreducible modulo p², it certainly cannot decompose in Z[x] into non-constant factors.

Subsequently, Eisenstein published a somewhat different version in 1850, also in Crelle's Journal. This version reads in translation

When in a polynomial F(x) in Template:Mvar of arbitrary degree the coefficient of the highest term is 1, and all following coefficients are whole (real, complex) numbers, into which a certain (real resp. complex) prime number Template:Mvar divides, and when furthermore the last coefficient is equal to εm, where ε denotes a number not divisible by Template:Mvar: then it is impossible to bring F(x) into the form

$$(x^{\mu }+a_1{x}^{\mu -1}+\cdots +a_{\mu })(x^{\nu }+b_1{x}^{\nu -1}+\cdots +b_{\nu })$$

where μ, ν ≥ 1, μ + ν = deg(F(x)), and all Template:Mvar and Template:Mvar are whole (real resp. complex) numbers; the equation F(x) = 0 is therefore irreducible.

Here "whole real numbers" are ordinary integers and "whole complex numbers" are Gaussian integers; one should similarly interpret "real and complex prime numbers". The application for which Eisenstein developed his criterion was establishing the irreducibility of certain polynomials with coefficients in the Gaussian integers that arise in the study of the division of the lemniscate into pieces of equal arc-length.

Remarkably Schönemann and Eisenstein, once having formulated their respective criteria for irreducibility, both immediately apply it to give an elementary proof of the irreducibility of the cyclotomic polynomials for prime numbers, a result that Gauss had obtained in his Disquisitiones Arithmeticae with a much more complicated proof. In fact, Eisenstein adds in a footnote that the only proof for this irreducibility known to him, other than that of Gauss, is one given by Kronecker in 1845. This shows that he was unaware of the two different proofs of this statement that Schönemann had given in his 1846 article, where the second proof was based on the above-mentioned criterion. This is all the more surprising given the fact that two pages further, Eisenstein actually refers (for a different matter) to the first part of Schönemann's article. In a note ("Notiz") that appeared in the following issue of the Journal, Schönemann points this out to Eisenstein, and indicates that the latter's method is not essentially different from the one he used in the second proof.

Basic proof

To prove the validity of the criterion, suppose Template:Mvar satisfies the criterion for the prime number Template:Mvar, but that it is nevertheless reducible in Q[x], from which we wish to obtain a contradiction. From Gauss' lemma it follows that Template:Mvar is reducible in Z[x] as well, and in fact can be written as the product Q = GH of two non-constant polynomials G, H (in case Template:Mvar is not primitive, one applies the lemma to the primitive polynomial Q/c (where the integer Template:Mvar is the content of Template:Mvar) to obtain a decomposition for it, and multiplies Template:Mvar into one of the factors to obtain a decomposition for Template:Mvar). Now reduce Q = GH modulo Template:Mvar to obtain a decomposition in (Z/pZ)[x]. But by hypothesis this reduction for Template:Mvar leaves its leading term, of the form axⁿ for a non-zero constant a ∈ Z/pZ, as the only nonzero term. But then necessarily the reductions modulo Template:Mvar of Template:Mvar and Template:Mvar also make all non-leading terms vanish (and cannot make their leading terms vanish), since no other decompositions of axⁿ are possible in (Z/pZ)[x], which is a unique factorization domain. In particular the constant terms of G and Template:Mvar vanish in the reduction, so they are divisible by Template:Mvar, but then the constant term of Template:Mvar, which is their product, is divisible by p², contrary to the hypothesis, and one has a contradiction.

A second proof of Eisenstein's criterion also starts with the assumption that the polynomial Q(x) is reducible. It is shown that this assumption entails a contradiction.

The assumption that $$Q(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$$ is reducible means that there are polynomials $$\begin{array}{rlrl}G(x)& =c_r{x}^r+c_{r-1}{x}^{r-1}+\cdots +c_0& & r\ge 1\\ H(x)& =d_s{x}^s+d_{s-1}{x}^{s-1}+\cdots +d_0& & s\ge 1\end{array}$$ Such that $$Q(x)=G(x)\cdot H(x),n=r+s.$$ The coefficient a₀ of the polynomial Q(x) can be divided by the prime Template:Mvar but not by p². Since a₀ = c₀d₀, it is possible to divide c₀ or d₀ by Template:Mvar, but not both. One may without loss of generality proceed

• with a coefficient c₀ that can be divided by Template:Mvar and
• with a coefficient d₀ that cannot be divided by Template:Mvar.

By the assumption, $p$ does not divide $a_n$. Because a_n = c_r d_s, neither c_r nor d_s can be divided by Template:Mvar. Thus, if $a_r$ is the $r$-th coefficient of the reducible polynomial $Q$, then (possibly with $d_t=0$ in case $t>s$) $$a_r=c_r{d}_0+c_{r-1}{d}_1+\cdots +c_0{d}_r$$ wherein $c_r{d}_0$ cannot be divided by $p$, because neither $d_0$ nor $c_r$ can be divided by $p$.

We will prove that $c_0,c_1,\dots ,c_{r-1}$ are all divisible by Template:Mvar. As $a_r$ is also divisible by Template:Mvar (by hypothesis of the criterion), this implies that $$c_r{d}_0=a_r-(c_{r-1}{d}_1+\cdots +c_0{d}_r)$$ is divisible by Template:Mvar, a contradiction proving the criterion.

It is possible to divide $c_0{d}_r$ by $p$, because $c_0$ can be divided by $p$.

By initial assumption, it is possible to divide the coefficient a₁ of the polynomial Q(x) by Template:Mvar. Since $$a_1=c_0{d}_1+c_1{d}_0$$ and since d₀ is not a multiple of Template:Mvar it must be possible to divide c₁ by Template:Mvar. Analogously, by induction, $c_i$ is a multiple of $p$ for all $i<r$, which finishes the proof.

Advanced explanation

Applying the theory of the Newton polygon for the [[p-adic number|Template:Mvar-adic number]] field, for an Eisenstein polynomial, we are supposed to take the lower convex envelope of the points

where v_i is the [[additive p-adic valuation|Template:Mvar-adic valuation]] of a_i (i.e. the highest power of Template:Mvar dividing it). Now the data we are given on the v_i for 0 < i < n, namely that they are at least one, is just what we need to conclude that the lower convex envelope is exactly the single line segment from (0, 1) to (n, 0), the slope being −1/n.

This tells us that each root of Template:Mvar has Template:Mvar-adic valuation 1/n and hence that Template:Mvar is irreducible over the Template:Mvar-adic field (since, for instance, no product of any proper subset of the roots has integer valuation); and a fortiori over the rational number field.

This argument is much more complicated than the direct argument by reduction mod Template:Mvar. It does however allow one to see, in terms of algebraic number theory, how frequently Eisenstein's criterion might apply, after some change of variable; and so limit severely the possible choices of Template:Mvar with respect to which the polynomial could have an Eisenstein translate (that is, become Eisenstein after an additive change of variables as in the case of the Template:Mvar-th cyclotomic polynomial).

In fact only primes Template:Mvar ramifying in the extension of Q generated by a root of Template:Mvar have any chance of working. These can be found in terms of the discriminant of Template:Mvar. For example, in the case x² + x + 2 given above, the discriminant is −7 so that 7 is the only prime that has a chance of making it satisfy the criterion. Modulo 7, it becomes (x − 3)²— a repeated root is inevitable, since the discriminant is 0 mod 7. Therefore, the variable shift is actually something predictable.

Again, for the cyclotomic polynomial, it becomes

the discriminant can be shown to be (up to sign) p^p−2, by linear algebra methods.

More precisely, only totally ramified primes have a chance of being Eisenstein primes for the polynomial. (In quadratic fields, ramification is always total, so the distinction is not seen in the quadratic case like x² + x + 2 above.) In fact, Eisenstein polynomials are directly linked to totally ramified primes, as follows: if a field extension of the rationals is generated by the root of a polynomial that is Eisenstein at Template:Mvar then Template:Mvar is totally ramified in the extension, and conversely if Template:Mvar is totally ramified in a number field then the field is generated by the root of an Eisenstein polynomial at Template:Mvar.

Generalization

Generalized criterion

Given an integral domain Template:Mvar, let $$Q=\sum _{i=0}^n{a}_i{x}^i$$ be an element of D[x], the polynomial ring with coefficients in Template:Mvar.

Suppose there exists a prime ideal p of Template:Mvar such that

• a_i ∈ p for each i ≠ n,
• a_n ∉ p, and
• a₀ ∉ p², where p² is the ideal product of p with itself.

Then Template:Mvar cannot be written as a product of two non-constant polynomials in D[x]. If in addition Template:Mvar is primitive (i.e., it has no non-trivial constant divisors), then it is irreducible in D[x]. If Template:Mvar is a unique factorization domain with field of fractions Template:Mvar, then by Gauss's lemma Template:Mvar is irreducible in F[x], whether or not it is primitive (since constant factors are invertible in F[x]); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of Template:Mvar. The latter statement gives original theorem for D = Z or (in Eisenstein's formulation) for D = Z[i].

Proof

The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo p; the essential point is that a single-term polynomial over the integral domain D/p cannot decompose as a product in which at least one of the factors has more than one term (because in such a product there can be no cancellation in the coefficient either of the highest or the lowest possible degree).

Example

After Z, one of the basic examples of an integral domain is the polynomial ring D = k[u] in the variable Template:Mvar over the field Template:Mvar. In this case, the principal ideal generated by Template:Mvar is a prime ideal. Eisenstein's criterion can then be used to prove the irreducibility of a polynomial such as Q(x) = x³ + ux + u in D[x]. Indeed, Template:Mvar does not divide a₃, u² does not divide a₀, and Template:Mvar divides a₀, a₁ and a₂. This shows that this polynomial satisfies the hypotheses of the generalization of Eisenstein's criterion for the prime ideal p = (u) since, for a principal ideal (u), being an element of (u) is equivalent to being divisible by Template:Mvar.

See also

• Cohn's irreducibility criterion
• Perron's irreducibility criterion

References

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