Eisenstein's criterion
Encyclopedia
In mathematics
, Eisenstein's criterion gives an easily checked sufficient condition
for a polynomial
with integer coefficients to be irreducible
over the rational number
s. The result is also known as the Schönemann–Eisenstein theorem; although this name is rarely used nowadays, it was common in the early 20th century.
Suppose we have the following polynomial with integer coefficients.
If there exists a prime number
p such that the following three conditions all apply:
then 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 as integer polynomial will have some prime number, necessarily distinct from p, as an irreducible factor). The latter possibility can be avoided by first making primitive, by dividing it by the greatest common divisor
of its coefficients (the contents of ). This division does not change whether is reducible or not over the rational numbers, and will not invalidate the hypotheses of the criterion for p (on the contrary it could make the criterion hold for some prime, even if it did not before the division).
This criterion is certainly not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but does allow in certain important particular cases to prove irreducibility with very little effort. In some cases the criterion does not apply directly (for any prime number), but it does apply after transformation of the polynomial, in such a way that irreducibility of the original polynomial can be concluded.
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 a) of for x; 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 , in which the coefficient 1 of x is not divisible by any prime, Eisenstein's criterion does not apply to H. But if one substitutes for x in H, one obtains the polynomial
, which satisfies Eisenstein's criterion for the prime number 7. Since the substitution is an automorphism
of the ring , 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 H (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 x anyway). This is so because such polynomials are reducible in if and only if they are reducible in (for any integral domain ), and in that ring the substitution of for 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 satisfies the criterion for after reversing its coefficients, and (being primitive) is therefore irreducible in .
s for prime numbers p. Such a polynomial is obtained by dividing the polynomial by the linear factor corresponding to its obvious root 1 (in fact its only rational root if ):
Here, as in the earlier example of , the coefficients 1 prevent Eisenstein's criterion from applying directly. However the polynomial will satisfy the criterion for p after substitution of for x: this gives
all of whose non-leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient equal to p, and therefore not divisible by p2. An alternative way to arrive at this conclusion is to use the identity which is valid in characteristic
p (and which is based on the same properties of binomial coefficients, and gives rise to the Frobenius endomorphism
), to compute the reduction modulo p of the quotient of polynomials:
which means that the non-leading coefficients of the quotient are all divisible by p; the remaining verification that the constant term of the quotient is p can be done by substituting 1 (instead of ) for x into the expanded form .
was the first to publish a version of the criterion, in 1846 in Crelle's Journal
, which reads in translation
This formulation already incorporates a shift to in place of 0; the condition on means that is not divisible by , and so is divisible by but not by . As stated it is not entirely correct in that it makes no assumptions on the degree of the polynomial , so that the polynomial considered need not be of the degree that its expression suggests; the example modulo shows the conclusion is not valid without such hypothesis. Assuming that the degree of does not exceed , the criterion is correct however, and somewhat stronger than the formulation given above, since if is irreducible modulo , it certainly cannot decompose in into non-constant factors.
Subsequently Eisenstein published a somewhat different version in in 1850, also in Crelle's Journal. This version reads in translation
Here "whole real numbers" are ordinary integer
s and "whole complex numbers" are Gaussian integer
s; 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 lemnsicate
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 two different proofs of this statement that Schönemann had given, one in either part of a two-part article, the second of which being the one based on the criterion cited above; 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.
it follows that is reducible in as well, and in fact can be written as the product of two non-constant polynomials , (in case is not primitive, one applies the lemma to the primitive polynomial (where the integer is the content of ) to obtain a decomposition for it, and multiplies c into one of the factors to obtain a decomposition for ). Now reduce modulo to obtain a decomposition in . But by hypothesis this reduction for leaves its leading term, of the form for a non-zero constant , as the only nonzero term. But then necessarily the reductions modulo of and also make all non-leading terms vanish (and cannot make their leading terms vanish), since no other decompositions of are possible in , which is a unique factorization domain
. In particular the constant terms of and vanish in the reduction, so they are divisible by , but then the constant term of , which is their product, is divisible by , contrary to the hypothesis, and one has a contradiction.
for the p-adic number
field, for an Eisenstein polynomial, we are supposed to take the lower convex envelope of the points
, (1, v1), (2, v2), ..., (n − 1, vn−1), (n,0),
where vi is the p-adic valuation of ai (i.e. the highest power of p dividing it). Now the data we are given on the vi 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 Q has p-adic valuation 1/n and hence that Q is irreducible over the p-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 p. 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 p 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 p-th cyclotomic polynomial).
In fact only primes p ramifying
in the extension of ℚ generated by a root of Q have any chance of working. These can be found in terms of the discriminant of Q. 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
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
p − 1 mod p;
the discriminant can be shown to be (up to sign) pp − 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 p then p is totally
ramified in the extension, and conversely if p is totally ramified in a number field then the field is generated by the root of
an Eisenstein polynomial at p.
be an element of , the polynomial ring
with coefficients in .
Suppose there exists a prime ideal
of such that
Then cannot be written as a product of two non-constant polynomials in . If in addition is primitive
(i.e., it has no non-trivial constant divisors), then it is irreducible in . If is a unique factorization domain
with field of fractions
, then by Gauss's lemma
is irreducible in , whether or not it is primitive (since constant factors are invertible in ); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of . The latter statement gives original theorem for or (in Eisenstein's formulation) for .
The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo
; the essential point is that a single-term polynomial over the integral domain 
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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Eisenstein
Necessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...
for a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
with integer coefficients to be irreducible
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
over the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. The result is also known as the Schönemann–Eisenstein theorem; although this name is rarely used nowadays, it was common in the early 20th century.
Suppose we have the following polynomial with integer coefficients.
If there exists a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p such that the following three conditions all apply:
- p divides each ai for i ≠ n,
- p does not divide an, and
- p2 does not divide a0,
then 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 as integer polynomial will have some prime number, necessarily distinct from p, as an irreducible factor). The latter possibility can be avoided by first making primitive, by dividing it by the greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
of its coefficients (the contents of ). This division does not change whether is reducible or not over the rational numbers, and will not invalidate the hypotheses of the criterion for p (on the contrary it could make the criterion hold for some prime, even if it did not before the division).
This criterion is certainly not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but does allow in certain important particular cases to prove irreducibility with very little effort. In some cases the criterion does not apply directly (for any prime number), but it does apply after transformation of the polynomial, in such a way that irreducibility of the original polynomial can be concluded.
Examples
Consider the polynomial . In order for Eisenstein's criterion to apply for a prime number p 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 is irreducible over the rational numbers (and since it is primitive, over the integers as well). Note that since is of degree 4, this conclusion could not have been established by only checking that has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible.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 a) of for x; 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 , in which the coefficient 1 of x is not divisible by any prime, Eisenstein's criterion does not apply to H. But if one substitutes for x in H, one obtains the polynomial
, which satisfies Eisenstein's criterion for the prime number 7. Since the substitution is an automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
of the ring , 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 H (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 x anyway). This is so because such polynomials are reducible in if and only if they are reducible in (for any integral domain ), and in that ring the substitution of for 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 satisfies the criterion for after reversing its coefficients, and (being primitive) is therefore irreducible in .
Cyclotomic polynomials
An important class of polynomials whose irreducibility can be established using Eisenstein's criterion is that of the cyclotomic polynomialCyclotomic polynomial
In algebra, the nth cyclotomic polynomial, for any positive integer n, is the monic polynomial:\Phi_n = \prod_\omega \,where the product is over all nth primitive roots of unity ω in a field, i.e...
s for prime numbers p. Such a polynomial is obtained by dividing the polynomial by the linear factor corresponding to its obvious root 1 (in fact its only rational root if ):
Here, as in the earlier example of , the coefficients 1 prevent Eisenstein's criterion from applying directly. However the polynomial will satisfy the criterion for p after substitution of for x: this gives
all of whose non-leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient equal to p, and therefore not divisible by p2. An alternative way to arrive at this conclusion is to use the identity which is valid in characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
p (and which is based on the same properties of binomial coefficients, and gives rise to the Frobenius endomorphism
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
), to compute the reduction modulo p of the quotient of polynomials:
which means that the non-leading coefficients of the quotient are all divisible by p; the remaining verification that the constant term of the quotient is p can be done by substituting 1 (instead of ) for x into the expanded form .
History
The criterion is named after Gotthold Eisenstein. However, Theodor SchönemannTheodor Schönemann
Theodor Schönemann, also written Schoenemann, was a German mathematician who obtained several important results in number theory concerning the theory of congruences, which can be found in several publications in Crelle's journal, volumes 17 to 40...
was the first to publish a version of the criterion, in 1846 in Crelle's Journal
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
, which reads in translation
That will be irreducible to the modulus when to the modulus does not contain a factor .
This formulation already incorporates a shift to in place of 0; the condition on means that is not divisible by , and so is divisible by but not by . As stated it is not entirely correct in that it makes no assumptions on the degree of the polynomial , so that the polynomial considered need not be of the degree that its expression suggests; the example modulo shows the conclusion is not valid without such hypothesis. Assuming that the degree of does not exceed , the criterion is correct however, and somewhat stronger than the formulation given above, since if is irreducible modulo , it certainly cannot decompose in into non-constant factors.
Subsequently Eisenstein published a somewhat different version in in 1850, also in Crelle's Journal. This version reads in translation
When in a polynomial in 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 divides, and when furthermore the last coefficient is = , where denotes a number not divisible by : then it is impossible to bring into the form
where and , = the degree of , and all and are whole (real resp. complex) numbers; the equation is therefore irreducible.
Here "whole real numbers" are ordinary integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s and "whole complex numbers" are Gaussian integer
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
s; 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 lemnsicate
Lemniscate of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is...
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
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
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
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
in 1845. This shows that he was unaware of two different proofs of this statement that Schönemann had given, one in either part of a two-part article, the second of which being the one based on the criterion cited above; 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 satisfies the criterion for the prime number , but that it is nevertheless reducible in , from which we wish to obtain a contradiction. From Gauss' lemmaGauss's lemma (polynomial)
In algebra, in the theory of polynomials , Gauss's lemma is either of two related statements about polynomials with integer coefficients:...
it follows that is reducible in as well, and in fact can be written as the product of two non-constant polynomials , (in case is not primitive, one applies the lemma to the primitive polynomial (where the integer is the content of ) to obtain a decomposition for it, and multiplies c into one of the factors to obtain a decomposition for ). Now reduce modulo to obtain a decomposition in . But by hypothesis this reduction for leaves its leading term, of the form for a non-zero constant , as the only nonzero term. But then necessarily the reductions modulo of and also make all non-leading terms vanish (and cannot make their leading terms vanish), since no other decompositions of are possible in , which is a unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
. In particular the constant terms of and vanish in the reduction, so they are divisible by , but then the constant term of , which is their product, is divisible by , contrary to the hypothesis, and one has a contradiction.
Advanced explanation
Applying the theory of the Newton polygonNewton polygon
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields.In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ringover K, where K...
for the p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
field, for an Eisenstein polynomial, we are supposed to take the lower convex envelope of the points
, (1, v1), (2, v2), ..., (n − 1, vn−1), (n,0),
where vi is the p-adic valuation of ai (i.e. the highest power of p dividing it). Now the data we are given on the vi 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
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
being −1/n.
This tells us that each root of Q has p-adic valuation 1/n and hence that Q is irreducible over the p-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 p. It does however allow one to see, in terms of algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, how frequently Eisenstein's criterion might apply, after some change of variable; and so limit severely the possible choices of p 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 p-th cyclotomic polynomial).
In fact only primes p ramifying
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
in the extension of ℚ generated by a root of Q have any chance of working. These can be found in terms of the discriminant of Q. 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
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
p − 1 mod p;
the discriminant can be shown to be (up to sign) pp − 2, by linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
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 p then p is totally
ramified in the extension, and conversely if p is totally ramified in a number field then the field is generated by the root of
an Eisenstein polynomial at p.
Generalization
Given an integral domain , let be an element of , the polynomial ringPolynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
with coefficients in .
Suppose there exists a prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
of such that- ai ∈
for each i ≠ n, - an ∉
, - a0 ∉
(where 
is the ideal product of 
with itself).
Then cannot be written as a product of two non-constant polynomials in . If in addition is primitive
Primitive polynomial
In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF...
(i.e., it has no non-trivial constant divisors), then it is irreducible in . If is a unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
with field of fractions
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
, then by Gauss's lemma
Gauss's lemma (polynomial)
In algebra, in the theory of polynomials , Gauss's lemma is either of two related statements about polynomials with integer coefficients:...
is irreducible in , whether or not it is primitive (since constant factors are invertible in ); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of . The latter statement gives original theorem for or (in Eisenstein's formulation) for .
The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo
; the essential point is that a single-term polynomial over the integral domain 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).