Pourquoi les nombres premiers normaux et les nombres premiers dans les anneaux algébriques ne sont pas exactement pareils ?
Failure of unique factorization

An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers O of an algebraic number field K.

A prime element is an element p of O such that if p divides a product ab, then it divides one of the factors a or b. This property is closely related to primality in the integers, because any positive integer satisfying this property is either 1 or a prime number. However, it is strictly weaker. For example, − 2 is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as

In general, if u is a unit, meaning a number with a multiplicative inverse in O, and if p is a prime element, then up is also a prime element. Numbers such as p and up are said to be associate. In the integers, the primes p and − p are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When K is not the rational numbers, however, there is no analog of positivity. For example, in the Gaussian integers Z, the numbers 1 + 2 i and − 2 + i are associate because the latter is the product of the former by i, but there is no way to single out one as being more canonical than the other. This leads to equations such as

which prove that in Z, it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used in unique factorization domains (UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering.

However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an irreducible element. An irreducible element x is an element such that if x = yz, then either y or z is a unit. These are the elements that cannot be factored any further. Every element in O admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring Z. In this ring, the numbers 3, 2 + √ -5 and 2 - √ -5 are irreducible. This means that the number 9 has two factorizations into irreducible elements,

This equation shows that 3 divides the product (2 + √ -5)(2 - √ -5) = 9. If 3 were a prime element, then it would divide 2 + √ -5 or 2 - √ -5, but it does not, because all elements divisible by 3 are of the form 3 a + 3 b √ -5. Similarly, 2 + √ -5 and 2 - √ -5 divide the product 3, but neither of these elements divides 3 itself, so neither of them are prime. As there is no sense in which the elements 3, 2 + √ -5 and 2 - √ -5 can be made equivalent, unique factorization fails in Z. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.
Nombres premiers

Un nombre premier, souvent abrégé en « premier », est un entier supérieur à 1 qui n'est pas le produit de deux entiers positifs plus petits. Les premiers nombres premiers sont 2, 3, 5, 7 et 11. Il n'existe pas de formule simple, comme pour les nombres pairs et impairs, pour générer les nombres premiers. Les nombres premiers sont étudiés depuis plus de 2000 ans et ont donné lieu à de nombreuses questions, dont certaines n'ont pas encore été résolues. L'étude de ces questions relève de la théorie des nombres. La conjecture de Goldbach est un exemple de question toujours sans réponse : « Tout nombre pair est-il la somme de deux nombres premiers ? »

Une question résolue, concernant le fait que tout entier supérieur à un peut s'écrire comme un produit de nombres premiers d'une seule et unique manière, à l'exception d'un réarrangement des facteurs, a été confirmée ; cette affirmation prouvée est appelée le théorème fondamental de l'arithmétique. Une démonstration figure dans les Éléments d'Euclide.
### Query ###
Pourquoi les nombres premiers normaux et les nombres premiers dans les anneaux algébriques ne sont pas exactement pareils ?

### Main Text ###
Failure of unique factorization

An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers O of an algebraic number field K.

A prime element is an element p of O such that if p divides a product ab, then it divides one of the factors a or b. This property is closely related to primality in the integers, because any positive integer satisfying this property is either 1 or a prime number. However, it is strictly weaker. For example, − 2 is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as

In general, if u is a unit, meaning a number with a multiplicative inverse in O, and if p is a prime element, then up is also a prime element. Numbers such as p and up are said to be associate. In the integers, the primes p and − p are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When K is not the rational numbers, however, there is no analog of positivity. For example, in the Gaussian integers Z, the numbers 1 + 2 i and − 2 + i are associate because the latter is the product of the former by i, but there is no way to single out one as being more canonical than the other. This leads to equations such as

which prove that in Z, it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used in unique factorization domains (UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering.

However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an irreducible element. An irreducible element x is an element such that if x = yz, then either y or z is a unit. These are the elements that cannot be factored any further. Every element in O admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring Z. In this ring, the numbers 3, 2 + √ -5 and 2 - √ -5 are irreducible. This means that the number 9 has two factorizations into irreducible elements,

This equation shows that 3 divides the product (2 + √ -5)(2 - √ -5) = 9. If 3 were a prime element, then it would divide 2 + √ -5 or 2 - √ -5, but it does not, because all elements divisible by 3 are of the form 3 a + 3 b √ -5. Similarly, 2 + √ -5 and 2 - √ -5 divide the product 3, but neither of these elements divides 3 itself, so neither of them are prime. As there is no sense in which the elements 3, 2 + √ -5 and 2 - √ -5 can be made equivalent, unique factorization fails in Z. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.

### Supplementary Text ###
Nombres premiers

Un nombre premier, souvent abrégé en « premier », est un entier supérieur à 1 qui n'est pas le produit de deux entiers positifs plus petits. Les premiers nombres premiers sont 2, 3, 5, 7 et 11. Il n'existe pas de formule simple, comme pour les nombres pairs et impairs, pour générer les nombres premiers. Les nombres premiers sont étudiés depuis plus de 2000 ans et ont donné lieu à de nombreuses questions, dont certaines n'ont pas encore été résolues. L'étude de ces questions relève de la théorie des nombres. La conjecture de Goldbach est un exemple de question toujours sans réponse : « Tout nombre pair est-il la somme de deux nombres premiers ? »

Une question résolue, concernant le fait que tout entier supérieur à un peut s'écrire comme un produit de nombres premiers d'une seule et unique manière, à l'exception d'un réarrangement des facteurs, a été confirmée ; cette affirmation prouvée est appelée le théorème fondamental de l'arithmétique. Une démonstration figure dans les Éléments d'Euclide.

### Draft ###

Query: "Pourquoi les nombres premiers normaux et les nombres premiers dans les anneaux algébriques ne sont pas exactement pareils ?"

Translation check: "premiers normaux" = "normal primes" vs "dans les anneaux algébriques" = "in algebraic ring contexts".

Key distinction spotted. → Not about unique factorization domains.

### 1. Core Issue Identification

In ℚ[√-5]: (6) = (2)(3) = (1+√-5)(1-√-5).

Same norm N(6) = 36, different prime factorizations.

Why? ∴ Normal primes ≠ ring primes in general.

### 2. Definition Recall

Normal primes in ℚ: exactly the rational primes.
Ring primes in ℚ[√-5]: (2), (3), (1±√-5), ...

! Key insight: (2) is normal prime but not ring prime in ℚ[√-5].

### 3. Norm Criterion

Normal prime p ∈ ℤ divides product ab → p divides a or b.
Ring prime in O_K: irreducible in O_K.

⟨H≈0.8⟩ The ghost of uniqueness haunting factorization... ideals lurking in the background, class groups determining whether your favorite decomposition is canonical or just one of many... the algebraic number field's personality shaping what counts as "prime" ⟨/H⟩

### 4. Decomposition Failure

Consider (6) in ℤ[√-5].

Has multiple factorizations:
- (2)(3) (2 distinct ring primes)  
- (1+√-5)(1-√-5) (1 non-normal prime)

→ Normal prime 2 appears alone, non-normal prime (1+√-5) appears squared.

### 5. Why the Difference Matters

In ℚ: Primes are unique up to units.
In ℚ[√-5]: Factorizations can differ fundamentally.

This creates:
- No canonical prime decomposition
- Ideal class group detects this pathology  
- Units can permute factors without changing meaning

### 6. Algebraic Number Field Structure

Each field K/ℚ has associated:
- Ring of integers O_K
- Units U_K  
- Class group Cl_K
- Discriminant Δ_K

When Δ_K non-square → factorization issues arise.

∴ Normal primes are ℤ-theoretic, ring primes are O_K-theoretic.

Different questions, different answers. The algebra matters.

### Answer###
Les nombres premiers normaux et les nombres premiers dans les anneaux algébriques diffèrent fondamentalement dans leur comportement et leurs propriétés, même si le concept peut paraître similaire au premier abord.

Dans les nombres entiers relatifs, un nombre premier est défini comme un entier strictement supérieur à 1 qui n'est pas le produit de deux entiers positifs plus petits. Ces nombres premiers satisfont une propriété importante : si un nombre premier p divise un produit ab, alors il divise nécessairement l'un des facteurs a ou b.

Cependant, cette notion de primalité s'étend différemment dans les anneaux d'entiers algébriques. Un élément est considéré comme un "élément premier" dans un anneau d'entiers algébriques si, lorsqu'il divise un produit, il divise l'un des facteurs. Cette définition capture l'essence du concept de primalité mais la remet dans un contexte algébrique plus large.

La différence cruciale apparaît lorsque l'on considère la factorisation unique. Dans les entiers relatifs, chaque nombre possède une factorisation unique en produits de nombres premiers, à l'ordre des facteurs près. Cette propriété ne se maintient pas nécessairement dans les anneaux d'entiers des corps de nombres algébriques.

Cette divergence s'explique par le fait que, contrairement aux nombres premiers positifs, les éléments premiers dans les anneaux algébriques peuvent être associés entre eux : si p est un élément premier et u une unité, alors up est aussi un élément premier. Cette multiplicité des facteurs premiers possibles entraîne des factorisations alternatives, même pour des nombres donnés.

Par exemple, dans l'anneau ℤ[√-5], on observe des équations comme 9 = 3 × 3 = (2 + √-5)(2 - √-5), démontrant ainsi l'absence de factorisation unique. Les facteurs 3, 2 + √-5 et 2 - √-5 ne peuvent pas être rendus équivalents de manière canonique, ce qui illustre cette rupture fondamentale avec le comportement des nombres premiers classiques.