Editing Prime number (section)

=== Prime ideals ===
{{Main|Prime ideals}}
Not every ring is a unique factorization domain. For instance, in the ring of numbers <math>a+b\sqrt{-5}</math> (for integers {{tmath|a}} and {{tmath|b}}) the number <math>21</math> has two factorizations {{tmath|1= 21=3\cdot7=(1+2\sqrt{-5})(1-2\sqrt{-5}) }}, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an [[ideal (ring theory)|ideal]], a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements.
''Prime ideals'', which generalize prime elements in the sense that the [[principal ideal]] generated by a prime element is a prime ideal, are an important tool and object of study in [[commutative algebra]], [[number theory|algebraic number theory]] and [[algebraic geometry]]. The prime ideals of the ring of integers are the ideals {{tmath|(0)}}, {{tmath|(2)}}, {{tmath|(3)}}, {{tmath|(5)}}, {{tmath|(7)}}, {{tmath|(11)}}, ... The fundamental theorem of arithmetic generalizes to the [[Lasker–Noether theorem]], which expresses every ideal in a [[Noetherian ring|Noetherian]] [[commutative ring]] as an intersection of [[primary ideal]]s, which are the appropriate generalizations of [[prime power]]s.<ref>{{cite book | last1=Eisenbud | first1=David | author1-link= David Eisenbud | title=Commutative Algebra | publisher=Springer-Verlag | location=Berlin; New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1| mr=1322960 | year=1995 | volume=150 | at=Section 3.3| doi=10.1007/978-1-4612-5350-1 }}</ref>

The [[spectrum of a ring]] is a geometric space whose points are the prime ideals of the ring.<ref>{{cite book | last = Shafarevich | first = Igor R. | author-link = Igor Shafarevich | doi = 10.1007/978-3-642-38010-5 | edition = 3rd | isbn = 978-3-642-38009-9 | mr = 3100288 | publisher = Springer, Heidelberg | title = Basic Algebraic Geometry 2: Schemes and Complex Manifolds | year = 2013 | contribution = Definition of <math>\operatorname{Spec} A</math> | page = 5 | contribution-url = https://books.google.com/books?id=zDW8BAAAQBAJ&pg=PA5}}</ref> [[Arithmetic geometry]] also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or [[Splitting of prime ideals in Galois extensions|ramification]] of prime ideals when lifted to an [[field extension|extension field]], a basic problem of algebraic number theory, bears some resemblance with [[ramified cover|ramification in geometry]]. These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in the [[ring of integers]] of [[quadratic number field]]s can be used in proving [[quadratic reciprocity]], a statement that concerns the existence of square roots modulo integer prime numbers.<ref>{{cite book | last = Neukirch | first = Jürgen | author-link = Jürgen Neukirch | doi = 10.1007/978-3-662-03983-0 | isbn = 978-3-540-65399-8 | location = Berlin | mr = 1697859 | publisher = Springer-Verlag | series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | title = Algebraic Number Theory | volume = 322 | year = 1999 | at = Section I.8, p. 50}}</ref> Early attempts to prove [[Fermat's Last Theorem]] led to [[Ernst Kummer|Kummer]]'s introduction of [[regular prime]]s, integer prime numbers connected with the failure of unique factorization in the [[cyclotomic field|cyclotomic integers]].<ref>{{harvnb|Neukirch|1999}}, Section I.7, p. 38</ref> The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by [[Chebotarev's density theorem]], which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.<ref>{{cite journal | last1 = Stevenhagen | first1 = P. | last2 = Lenstra | first2 = H.W. Jr. | author2-link = Hendrik Lenstra | doi = 10.1007/BF03027290 | issue = 2 | journal = [[The Mathematical Intelligencer]] | mr = 1395088 | pages = 26–37 | title = Chebotarëv and his density theorem | volume = 18 | year = 1996| citeseerx = 10.1.1.116.9409 | s2cid = 14089091 }}</ref>