On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute Values

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over  . A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension L ¼ K α ð Þ , we give a method to describe all absolute values of L extending ∣∣ , where K , jj ð Þ is a discrete rank one valued field.


Introduction
Polynomial factorization over a field is very useful in algebraic number theory, for prime ideal factorization.It is also important in extensions of valuations, etc.For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors (cf.[1][2][3][4][5][6][7]).In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over  [1].A criterion which was generalized in 1906 by Dumas in [8], who showed that for a polynomial Þν p a 0 ð Þ> 0 for every 0 ¼ i, … , n À 1, and gcd ν p a 0 ð Þ, n À Á ¼ 1 for some prime integer p, then f x ð Þ is irreducible over .In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion [9].He showed for a valued field K, ν ð Þand for a monic polynomial where R ν is a valuation ring of a discrete rank one valuation and ϕ being a monic polynomial in R ν x ½ whose reduction ϕ is irreducible over Þν a 0 ð Þ for every i ¼ 0, … , n À 1 and gcd ν a 0 ð Þ, n ð Þ¼1, then f x ð Þ is irreducible over the field K.In this paper, based on absolute value, we give an irreduciblity criterion of monic polynomials.More precisely, let K, jj ð Þbe a discrete rank one valued field, R | | its valuation ring,  | | , its residue field, and Γ ¼ |K * | its value group, we show that for a monic and n is the smallest integer satisfying γ n ∈ Γ, where γ ¼ a 0 j j ∞ À Á 1=n , then f x ð Þ is irreducible over K. Similarly for the results of extensions of valuations given in [10,11], for any simple algebraic extension L ¼ K α ð Þ, we give a method to describe all absolute values of L extending ||, where K, jj ð Þis a discrete rank one valued field.Our results are illustrated by some examples.

Newton polygons
Let L ¼  α ð Þ be a number field generated by a complex root α of a monic irreducible polynomial f x ð Þ ∈  x ½ and  L the ring of integers of L. In 1894, K. Hensel developed a powerful approach by showing that the prime ideals of  L lying above a prime p are in one-one correspondence with monic irreducible factors of f x ð Þ in  p x ½ .For every prime ideal corresponding to any irreducible factor in  p x ½ , the ramification index and the residue degree together are the same as those of the local field defined by the irreducible factors [6].These results were generalized in ( [12], Proposition 8.2).Namely, for a rank one valued field K, ν ð Þ, R ν its valuation ring, and Þwill be defined later.So, in order to describe all valuations of L extending ν, one needs to factorize the polynomial f x ð Þ into monic irreducible factors over K h .The first step of the factorization was based on Hensel's lemma.Unfortunately, the factors provided by Hensel's lemma are not necessarily irreducible over K h .The Newton polygon techniques could refine the factorization.Namely, theorem of the product, theorem of the polygon, and theorem of residual polynomial say that we can factorize any factor provided by Hensel's lemma, with as many sides of the polygon and with as many of irreducible factors of the residual polynomial.For more details, we refer to [7,13] for Newton polygons over p-adic numbers and [14,15] for Newton polygons over rank one discrete valued fields.As our proofs are based on Newton polygon techniques, we recall some fundamental notations and techniques on Newton polygons.Let K, ν ð Þbe a rank one discrete valued field (ν K * ð Þ ¼ ), R ν its valuation ring, M ν its maximal ideal,  ν its residue field, and K h , ν h À Á its henselization; the separable closure of K in K, where K is the completion of K, jj ð Þ, and || is an associated absolute value of ν.By normalization, we can assume that ν

Upon the Euclidean division by successive powers of ϕ, we can expand
g in the Euclidean plane.For every edge S j , of the polygon N ϕ f ð Þ, let l j be the length of the projection of S j to the x-axis and H j the length of its projection to the y-axis.l j is called the length of S j and H j is its height.Let d j ¼ gcd l j , H j À Á be the degree of S j , e j ¼ l j d j the ramification degree of S j , and Àλ j ¼ À H j l j ∈  the slope of S j .Geometrically, we can remark that N ϕ f ð Þ is the process of joining the obtained edges S 1 , … , S r ordered by increasing slopes, which can be expressed by The segments S 1 , … , and S r are called the sides of ð Þ, which is determined by joining all sides of negative slopes.For every side S of the polygon N þ ϕ f ð Þ of slope Àλ and initial point s, u s ð Þ, let l be its length, H its height and e the smallest positive integer satisfying eλ ∈ .Since lλ ¼ H ∈ , we conclude that e divides l, and so d ¼ l=e ∈  called the degree of S.
For every i ¼ 0, … , l, we attach the following residue coefficient c i ∈  ϕ : Þlies on S: where π, ϕ ð Þ is the maximal ideal of R ν x ½ generated by π and ϕ.Let λ ¼ Àh=e be the slope of S, where h ¼ H=d and d ¼ l=e.Notice that, the points with integer coordinates lying in S are exactly s, Þdoes not lie on S, and so c i ¼ 0. It follows that the candidate abscissas which yield nonzero residue coefficient are s, s þ e, … , and Otherwise, T is reduced to a single point; the end point of a side S i , which is also the initial point of S iþ1 if λ iþ1 < λ < λ i or the initial point of N ϕ f ð Þif λ i < λ for every side S i of N ϕ f ð Þor the end point of N ϕ f ð Þif λ i < λ for every side S i of N ϕ f ð Þ.In the sequel, we denote by The following are the relevant theorems from Newton polygon.Namely, theorem of the product and theorem of the polygon.For more details, we refer to [15].
Theorem 2.1.(theorem of the product

Absolute values
Let || be an absolute value of K; a map || : K ! þ , which satisfies the following three axioms: 1. Show that if K is a finite field, then || is a non archimidean absolute value.

Let p be a prime integer and jj
, where ν p is the p-adic valuation on .Show that jj p is a non archimidean absolute value of .

Characteristic elements of an absolute value
Let K, jj ð Þbe a non archimedian valued field. Let

Completion and henselization
Let K, jj ð Þbe a valued field and consider the map d : Then d is a metric on K. Definition 1.A sequence u n ð Þ∈ K  is said to be a Cauchy sequence if for every positive real number ε, there exists an integer N such that for every natural numbers m, n ≥ N, we have Any convergente sequence of K, jj ð Þis a Cauchy sequence.The converse is false, indeed, it suffices to consider the valued field , jj 0 À Á with jj 0 is the usual absolute value of  and Þis a complete valued field valued field, then there exists The following example shows that for any prime integer p, Hensel's lemma is not applicable in , jj ð Þ, with || is the p-adic absolute value defined by |a| ¼ p Àν p a ð Þ .Indeed, let q be a prime integer which is coprime to p, n ≥ 2 an integer, and

Main results
Let K, jj ð Þbe a non archimidean valued field, ν the associated valuation to || defined by ν a ð Þ ¼ ÀLn|a| for every a ∈ K * , R | | its valuation ring, M | | its maximal ideal,  | | its residue field, and K h , ν h À Á its henselization.

Key polynomials
The notion of key polynomials was introduced in 1936, by MacLane [17], in the case of discrete rank one absolute values and developed in [18] by Vaquié to any arbitrary rank valuation.The motivation of introducing key polynomials was the problem of describing all extensions of || to any finite simple extension K α ð Þ.For any simple algebraic extension of K, MacLane introduced the notions of key polynomials and augmented absolute with respect to the gievn key.
Definition 7. Two nonzero polynomials f and 2. We say that g is ||-divides f if there exists q ∈ R | | x ½ such that f and gq are ||-equivalent.

We say that a polynomial
½ is said to be a MacLane-Vaquié key polynomial of || if it satisfies the following three conditions: It is easy to prove the following lemma:

Augmented absolute values
Then by the ultra-metric propriety For the equality, by definition of i 1 and i 2 , Thus by using the expression of c i 1 þi 2 , we conclude the equality.For

Extensions of absolute values
The following Lemma makes a one-one correspondence between the absolute value of L and monic irreducible factors of ð Þ be the factorization into powers of monic irreducible factors of K h x ½ .Then e i ¼ 1 for every i ¼ 1, … , t and there are exactly t distinct valuations jj 1 , … , and jj t of L extending ||.Furthermore for every absolute value jj i of L associated to the Then for every absolute value jj L of L extending ||, for every nonzero polynomial P ∈ K x ½ , P α ð Þ j j L ≤ P j j ∞ .The equality holds if and only if ϕ does not divide P 0 , where P 0 ¼ P a , with a ∈ K such that P In particular, since ½ be a nonzero polynomial of degree less than degree of ϕ.
As degree P 0 is less than degree of ϕ, ϕ does not divide P 0 .Thus The first point of the theorem is an immediate consequence of Theorem 2.6.For the second point, let By the previous case, we conclude that |b tÀi | ≤ τ i for every i ¼ 0, … , t with τ ¼ b 0 j j 1=t , which means that ð Þ has a single side of the same slope Àλ.Therefore, a nÀi j j ∞ ≤ γ i for every i ¼ 0, … , n, where γ ¼ a 0 j j 1=n ∞ .□ Exercices 6.Let K, jj ð Þbe a non archimidean valued field and Based on absolute value, the following theorem gives an hyper bound of the number of monic irreducible factors of monic polynomials.In particular, Corollary 3.9 gives a criterion to test the irreducibility of monic polynomials. Theorem Since || is a non archimidean absolute value, we conclude that jj L is a non archimidean absolute value, and so by the ultra-metric propriety, Now, let P ¼ P l i¼0 p i ϕ i be a polynomial in K x ½ .By the ultra-metric propriety, If n is the smallest positive integer satisfying γ e ∈ Γ, then there is a unique absolute value jj L of L extending ||.Moreover this absolute value is Let us show the equality.Let s be the smallest integer which satisfies ω For the residue degree and ramification index, since ϕ α , we conclude that n divides the ramification index e of jj L .On the other hand, since As m Á n ¼ deg f ð Þ, we conclude the equality.□ Exercices 7.For every positive integer n ≥ 2 and p a positive prime integer, let 1. How that f is irreducible over .
3. Let L ¼  α ð Þ with α a complex root of f x ð Þ. Show that there is a unique absolute value of L extending the absolute jj p , defined on  by a j j p ¼ e Àν p a ð Þ for every a ∈ , where ν p is the p-adic valuation on .Calculate its residue degree and its ramification index.
Combining Lemma 3.4 and Theorem 3.8, we conclude the following result: Corollary 3.12.Let L ¼ K α ð Þ be a simple extension generated by α ∈ K a root of a monic irreducible polynomial f  x i with l < n.

□ Definition 9 . 3 .
The absolute value ω defined in Theorem 3.3 is denoted by jjϕ, γ, ½ and called the augmented absolue value of || associated to ϕ and γ.Example Let |∥ be the 2-adic absolute value defined on  by |a| ¼ e Àν 2 a ð Þ , where for every integer b, and only if a 6 ¼ 0, 2. |ab| ¼ |akb|, and 3. |a þ b| ≤ |a| þ |b|.ðtriangular inequalityÞ for every a, b ð Þ∈ K 2 .If the triangular inequality is replaced by an ultra-inequality, namely |a þ b| ≤ max jaj, jbj f gfor every a, b ð Þ∈ K 2 , then the absolute value || is called a non archimidean absolute value and we say that K, jj ð Þis a non archimidean valued field.
Lemma 2.4.Let K, jj ð Þbe a valued field.Then || is a non archimidean absolute value if and only if the set jn1 K j, n ∈  f gis bounded in .Proof.By induction if || is a non archimidean absolute value, then the set jn1 K j, n ∈  f gis bounded by 1. Conversely, assume that there exists the residue field of ||.Exercices 2. Let p be a prime integer and || the p-adic absolute value of , defined by |a| ¼ p Àν p a ð Þ for every a ∈ , where ν p a ð Þ is the greatest integer satisfying p ν p a ð Þ divides a for a 6 ¼ 0 and ν p 0 ð Þ ¼ ∞.
[16]hy sequence, which is not convergente.Þbe a valued field extension and Δ ¼ L * j j L .Then e ¼ jΔ=Γj the cardinal order of Δ=Γ, is called the ramification index of the extension andf ¼  j j L : F | | Â Ãis called its residue degree.Show that the convergence of a series in K is equivalent to the convergence of its general term to 0.3.LetΓ ¼ |K * | and Δ ¼ L * j j L .Show that if Γ is a discrete rank one Abelian group and L=K is a finite extension, then Δ is a discrete rank one Abelian group andL : K ½ ¼ef, where e is the ramification index of the extension and f is its residue degree.Furtheremore, the completion is unique up to a valued fields isomorphism.Now we come to an important property of complete fields.This theorem is widely known as Hensel's Lemma.For the proof, we refer to ([16], Lemma 4.1.3).
Definition 3. Let K, jj ð Þbe a valued field, L=K an extension of fields, and jj L an absolute value of L.1.We say that jj L extends || if jj L and || coincide on K.In this case L, jj L [16], we conclude that Hensel's lemma is not applicable in , jj Show that K ⊂ K h ⊂ K. Furthermore, these three fields have the same value group and same residue fields.We have the following apparently easier characterization of Henselian fields.For the proof, we refer to ([16], Lemma 4.1.1).
s , where K s is the separable closure of K.In particular, we conclude the following characterization of the henselization K h of K, jj ð Þ. Theorem 2.8.Let K, jj ð Þbe a valued field.Then K h is the separable closure of K in K s .6RecentAdvances in Polynomials and assume that a nÀi j j ∞ ≤ γ i for every i ¼ 0, … , n, where γ ¼ a 0 j j 1=n ∞ .Let e be the smallest positive integer satisfying γ e ∈ Γ.By applying the map ÀLn, the hypothesis a nÀi j j ∞ ≤ γ i for every i ¼ 0, … , n Àλ is the slope of S i .Since e is the smallest positive integer satisfying γ e ∈ Γ, we conclude that e is the smallest positive integer satisfying eλ ∈ ν K * ð Þ.On the other hand, since λ ¼ a i0 l i is the slope of S i , where l i is the length of the side S i , we conclude that e divides l i .Thus deg f i If n ¼ e, then d ¼ 1, and so there is a unique monique polynomial of K x ½ which divides f x ð Þ and this factor has the degree at least mn.As deg f By Corollary 3.9, if n ¼ e, then f x ð Þ is irreducible over K h .Thus by Hensel's Lemma, there is a unique absolute value jj L of L extending ||.By Theorem 3.10, we conclude that ϕ ij is the degree of S ij , l ij is the length of S ij , and e ij ¼ l ij d ij for every i ¼ 1, … , r and j ¼ 1, … , g i .
S ig i be the principal ϕ i -Newton polygon of f x ð Þ.Then L has t absolue value extending || with r ≤ t ≤ e

4. Applications 1
. Let ∥ be the p-adic absolute value defined on  by |a| ¼ p Àν p a ð Þ and fx ð Þ ¼ x n À p x ½ .Show that f x ð Þ is irreducible over .Let L ¼  α ð Þ with α a complex root of f x ð Þ. Determine all absolute value of L extending ||.Answer.First Γ ¼ p k , k ∈  È É is the value group of ||.Since |p| ¼ p À1 , γ ¼ a 0 j j 1=n ¼ p À1=n, we conclude that the smallest integer satisfying γ e ∈ Γ is n.Thus, by Corollary 3.9, f x ð Þ is irreducible over  h ,and so is over.Since f x ð Þ ¼ x n in  p x½ , by Theorem 3.11, there is a unique absolute value of L extending || and it is defined byP α ð Þ j j L ¼ max f|p i |γ i , i ¼ o, … , lg for every polynomial P ¼ P l i¼0 x i with l < n.2.Let ∥ be the p-adic absolute value andf x ð Þ ¼ x n Àa ∈  x ½ such that p does not divide ν p a ð Þ. Show that f x ð Þ is irreducible over .Let L ¼  α ð Þ with α a complex root of f x ð Þ. Determine all absolute value of L extending ||.Answer.First Γ ¼ p k , k ∈  È Éis the value group of ||.Since |p| ¼ p À1 , γ ¼ a 0 j j 1=n ¼ p À1=n , we conclude that the smallest integer satisfying γ e ∈ Γ is n.Thus, by Corollary 3.9, f x ð Þ is irreducible over  h ,and so is over .Since f x ð Þ ¼ x n in  p x ½ , by Theorem 3.11, there is a unique absolute value of L extending || and it is defined by P α ð Þ j j L ¼ max f|p i |γ i , i ¼ o, … , lg for every polynomial P ¼ P l i¼0 48 with ϕ ∈  x ½ a monic polynomial whose reduction is irreducible in  2 x ½ .In  2 x ½ , how many monic irreducible factors f x ð Þ gets?, where  2 is the completion of , jj ð Þand || is the 2-adic absolute value.Answer.It is easy to check that f x ð Þ satisfies the conditions of Theorem 3.8;a 6Ài j j ∞ ≤ γ i with γ ¼ 2 À4=6 ¼ 2 À1=3 À Á 2 .Thus e ¼ 3 and d ¼ 2. By Theorem 3.8, f x ð Þ has at most 2 monic irreducible factors in  2 x ½ .