Unique factorization/Advanced: Difference between revisions
imported>Andrey Khalyavin (→Proof) |
imported>Barry R. Smith (number systems with UF) |
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We can give a more precise version as follows. The first part of the theorem states that every nonzero integer ''n'' has a prime factorization. We can therefore write | We can give a more precise version as follows. The first part of the theorem states that every nonzero integer ''n'' has a prime factorization. We can therefore write | ||
: <math> n = (-1)^{\varepsilon} \prod_{p} p^{e_p} | : <math> n = (-1)^{\varepsilon} \prod_{p} p^{e_p}, </math> | ||
this being an infinite product over the set of prime numbers. As ''n'' can have only finitely many prime factors, all but finitely many of the exponents <math> e_p </math> are 0, so the product makes sense. Also, we can restrict <math>\varepsilon </math> to be either 0 or 1 when ''n'' is positive or negative respectively, and all of the exponents <math> e_p </math> must be nonnegative integers. With these conventions, the fundamental theorem can now be precisely expressed as saying that for any nonzero integer ''n'', a factorization as above exists, and the list of integers <math> \left( \varepsilon, e_{2}, e_{3}, e_{5}, \ldots \right) </math> is uniquely determined by ''n''. | this being an infinite product over the set of prime numbers. As ''n'' can have only finitely many prime factors, all but finitely many of the exponents <math> e_p </math> are 0, so the product makes sense. Also, we can restrict <math>\varepsilon </math> to be either 0 or 1 when ''n'' is positive or negative respectively, and all of the exponents <math> e_p </math> must be nonnegative integers. With these conventions, the fundamental theorem can now be precisely expressed as saying that for any nonzero integer ''n'', a factorization as above exists, and the list of integers <math> \left( \varepsilon, e_{2}, e_{3}, e_{5}, \ldots \right) </math> is uniquely determined by ''n''. | ||
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== Proof == | == Proof == | ||
The proof of both the existence and uniqueness statements of the fundamental theorem requires the technique known as [[mathematical induction]]. | The proof of both the existence and uniqueness statements of the fundamental theorem requires the technique known as [[mathematical induction]]. The statement and proof can be easily generalized to [[principal ideal domain]]s. | ||
=== Existence === | |||
To see that every integer > 1 can be factored as the product of prime numbers, we begin the strong induction proof by observing that the first such integer, 2, is a product of primes (note: a single prime must itself be considered to be such a "product", or else the theorem would be false). We now fix an integer ''N'' and make the induction hypothesis by assuming that every integer between 2 and ''N''-1 inclusive is already known to factor into the product of prime numbers. Upon considering a possible factorization of ''N'', there are two cases: ''N'' is either prime or composite. If ''N'' is prime, then ''N'' by itself is a factorization into primes. Otherwise, ''N'' is composite. In other words, there exist integers ''m, n''>1 such that <math> N=mn </math>. Thus, <math> m=N/n </math>, so <math> m \leq N-1 </math> since ''n'' > 1. Similarly, <math> n \leq N-1 </math> since ''m'' > 1. Thus, both of the integers ''m'' and ''n'' fall into the range of integers covered by the induction hypothesis, so they can both be written as products of prime numbers. But then <math> N = mn </math> can also be written as the product of prime numbers. By the principle of strong induction, it follows that every integer > 1 can be written as the product of prime numbers. | |||
=== Uniqueness === | |||
Every integer <math>N > 1</math> can be written in a unique way as a product of prime factors, up to reordering. To see why this is true, assume that <math>N</math> can be written as a product of prime factors in two ways | Every integer <math>N > 1</math> can be written in a unique way as a product of prime factors, up to reordering. To see why this is true, assume that <math>N</math> can be written as a product of prime factors in two ways | ||
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Continuing in this way, we obtain a sequence of numbers <math>N = N_0 > N_1 > N_2 > \cdots > N_n = 1</math> where each <math>N_{\alpha}</math> is obtained by dividing <math>N_{\alpha - 1}</math> by a prime factor. In particular, we see that <math>m = n</math> and that there is some permutation ("rearrangement") σ of the indices <math>1, 2, \ldots n</math> such that <math>p_i = q_{\sigma(i)}</math>. Said differently, the two factorizations of ''N'' must be the same up to a possible rearrangement of terms. | Continuing in this way, we obtain a sequence of numbers <math>N = N_0 > N_1 > N_2 > \cdots > N_n = 1</math> where each <math>N_{\alpha}</math> is obtained by dividing <math>N_{\alpha - 1}</math> by a prime factor. In particular, we see that <math>m = n</math> and that there is some permutation ("rearrangement") σ of the indices <math>1, 2, \ldots n</math> such that <math>p_i = q_{\sigma(i)}</math>. Said differently, the two factorizations of ''N'' must be the same up to a possible rearrangement of terms. | ||
== A number system where unique factorization fails == | |||
There are other systems of numbers that have striking similarities with the integers. One example is the [[Gaussian integer]]s, <math> \mathbb{Z} [\sqrt{-1}] </math>. A Gaussian integer is built up from the usual (or rational) integers and the extra number <math> i </math>, a square root of -1, through a finite number of additions, subtractions, and multiplications. We may think of the Gaussian integers as the smallest system of numbers that you can obtain if you want to throw <math>i</math> in with the rational integers. After simplification, a Gaussian integer can always be written in the form <math> a + b i </math> where ''a'' and ''b'' are rational integers. There is the natural notion of a prime Gaussian integer, where such [[Gaussian prime]]s cannot be factored further in a certain sense. Also, there is a unique factorization theorem for the Gaussian integers, stating that each Gaussian integer can be factored as a product of Gaussian primes in an essentially unique way. | |||
Surprisingly, there are other similar number systems where unique factorization does not hold. The classic example is <math> \mathbb{Z} [ \sqrt{-5} ] </math>. A number in this system can always be written as <math> a + b \sqrt{-5} </math> for some rational integers ''a'' and ''b''. Once again, there are numbers in this system that can be considered to be 'atomic', in that the do not factor in any nontrivial way. In this context, such numbers are called [[irreducible]], not prime, because in examples where unique factorization does not hold, the word 'prime' is reserved for a different use. A unique factorization theorem would state that every number in <math> \mathbb{Z} [\sqrt{-5} ] </math> can be factored as a product of irreducible numbers in an essentially unique way. Essentially here means that the factorization is unique up to reordering the factors and multiplying by numbers dividing 1. It can be shown that the only numbers dividing 1 in this system are -1 and 1. | |||
Consider the following factorizations of 6 in <math> \mathbb{Z} [\sqrt{-5}] </math>: | |||
: <math> 6 = 2 \cdot 3 = (1 + \sqrt{-5}) (1 - \sqrt{-5}). </math> | |||
It turns out that 2, 3, and the numbers <math> 1 \pm \sqrt{-5} </math> are each irreducible. As 1 and -1 are the only numbers dividing 1 in this system, these are two ''essentially different'' factorizations of 6 into irreducible numbers. | |||
To see that the above numbers are irreducible, we need a tool, the [[norm (field)|field norm]], which is the function from <math> \mathbb{Z} [\sqrt{-5}] </math> to the integers given by | |||
: <math> N (a + b \sqrt{-5}) = a^2 + 5 b^2.</math> | |||
'''Give properties, proof of UF here''' | |||
== Number systems with unique factorization == | |||
As mentioned above, unique factorization holds in <math> \mathbb{Z} [\sqrt{-1}] </math>, but in <math> \mathbb{Z} [\sqrt{-5}] </math>, numbers do not have unique factorizations into irreducible factors. It was the search for a remedy for this that led, first to the invention of [[ideal number]]s by [[Ernst Eduard Kummer|Kummer]], and later to the development of [[ideal (mathematics)|ideal]]s and [[ring (mathematics)|ring]] theory by [[Julius Wilhelm Richard Dedekind|Dedekind]]. Eventually, unique factorization in this setting was salvaged by proving that the [[principal ideal]] generated by a number (and actually every ideal) has a unique factorization into [[prime ideal]]s. For more on this, read about [[Dedekind domain]]s. | |||
A ring is a mathematical structure similar to the integers, of which the integers, <math> \mathbb{Z} [\sqrt{-1}] </math>, and <math> \mathbb{Z} [\sqrt{-5}] </math> are all examples. Rings are the appropriate mathematical setting in which to consider the concepts of irreducible factors and unique factorization. Those rings in which unique factorization holds are called [[unique factorization domain]]s. | |||
Latest revision as of 05:25, 23 April 2008
In mathematics, the unique factorization theorem, also known as the fundamental theorem of arithmetic states that every nonzero integer can be written in a unique way as a product of a unit (i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm 1 } ) and prime numbers. Unique factorization is the foundation for most of the structure of whole numbers as described by elementary number theory. The formulation of many results (for instance, the Chinese remainder theorem) would either be nonsensical, or at least more complicated, if unique factorization did not hold.
Precise formulation
The theorem has two parts: first, that a factorization exsits, and second, that it is unique. This description leads to a couple of troublesome questions. If we allow rearrangement of the factors, as in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 = 2 \times 3 = 3 \times 2} , the theorem is false. Also, what are the prime factorizations of 1 and -1?
We can give a more precise version as follows. The first part of the theorem states that every nonzero integer n has a prime factorization. We can therefore write
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = (-1)^{\varepsilon} \prod_{p} p^{e_p}, }
this being an infinite product over the set of prime numbers. As n can have only finitely many prime factors, all but finitely many of the exponents Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_p } are 0, so the product makes sense. Also, we can restrict Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon } to be either 0 or 1 when n is positive or negative respectively, and all of the exponents Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_p } must be nonnegative integers. With these conventions, the fundamental theorem can now be precisely expressed as saying that for any nonzero integer n, a factorization as above exists, and the list of integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \varepsilon, e_{2}, e_{3}, e_{5}, \ldots \right) } is uniquely determined by n.
For a nonnegative integer n and a prime number p, the exponent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_{p} } in the factorization is called the order of n at p, written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ord}_p (n) } . Consideration of these orders leads directly to the useful theory of valuations.
Proof
The proof of both the existence and uniqueness statements of the fundamental theorem requires the technique known as mathematical induction. The statement and proof can be easily generalized to principal ideal domains.
Existence
To see that every integer > 1 can be factored as the product of prime numbers, we begin the strong induction proof by observing that the first such integer, 2, is a product of primes (note: a single prime must itself be considered to be such a "product", or else the theorem would be false). We now fix an integer N and make the induction hypothesis by assuming that every integer between 2 and N-1 inclusive is already known to factor into the product of prime numbers. Upon considering a possible factorization of N, there are two cases: N is either prime or composite. If N is prime, then N by itself is a factorization into primes. Otherwise, N is composite. In other words, there exist integers m, n>1 such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=mn } . Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=N/n } , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \leq N-1 } since n > 1. Similarly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \leq N-1 } since m > 1. Thus, both of the integers m and n fall into the range of integers covered by the induction hypothesis, so they can both be written as products of prime numbers. But then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = mn } can also be written as the product of prime numbers. By the principle of strong induction, it follows that every integer > 1 can be written as the product of prime numbers.
Uniqueness
Every integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N > 1} can be written in a unique way as a product of prime factors, up to reordering. To see why this is true, assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} can be written as a product of prime factors in two ways
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = p_1 p_2 \cdots p_m = q_1 q_2 \cdots q_n}
We may now use a technique known as mathematical induction to show that the two prime decompositions are really the same.
Consider the prime factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1} . We know that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1 \mid q_1 q_2 \cdots q_n.}
Using the second definition of prime numbers, it follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1} divides one of the q-factors, say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_i} . Using the first definition, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1} is in fact equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_i}
Now, if we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1 = N/p_1} , we may write
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1 = p_2 p_3 \cdots p_m}
and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1 = q_1 q_2 \cdots q_{i-1} q_{i+1} \cdots q_n}
In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1} is the product of all the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} 's except Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_i} .
Continuing in this way, we obtain a sequence of numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = N_0 > N_1 > N_2 > \cdots > N_n = 1} where each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{\alpha}} is obtained by dividing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{\alpha - 1}} by a prime factor. In particular, we see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = n} and that there is some permutation ("rearrangement") σ of the indices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1, 2, \ldots n} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i = q_{\sigma(i)}} . Said differently, the two factorizations of N must be the same up to a possible rearrangement of terms.
A number system where unique factorization fails
There are other systems of numbers that have striking similarities with the integers. One example is the Gaussian integers, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [\sqrt{-1}] } . A Gaussian integer is built up from the usual (or rational) integers and the extra number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } , a square root of -1, through a finite number of additions, subtractions, and multiplications. We may think of the Gaussian integers as the smallest system of numbers that you can obtain if you want to throw Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} in with the rational integers. After simplification, a Gaussian integer can always be written in the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + b i } where a and b are rational integers. There is the natural notion of a prime Gaussian integer, where such Gaussian primes cannot be factored further in a certain sense. Also, there is a unique factorization theorem for the Gaussian integers, stating that each Gaussian integer can be factored as a product of Gaussian primes in an essentially unique way.
Surprisingly, there are other similar number systems where unique factorization does not hold. The classic example is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [ \sqrt{-5} ] } . A number in this system can always be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + b \sqrt{-5} } for some rational integers a and b. Once again, there are numbers in this system that can be considered to be 'atomic', in that the do not factor in any nontrivial way. In this context, such numbers are called irreducible, not prime, because in examples where unique factorization does not hold, the word 'prime' is reserved for a different use. A unique factorization theorem would state that every number in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [\sqrt{-5} ] } can be factored as a product of irreducible numbers in an essentially unique way. Essentially here means that the factorization is unique up to reordering the factors and multiplying by numbers dividing 1. It can be shown that the only numbers dividing 1 in this system are -1 and 1.
Consider the following factorizations of 6 in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [\sqrt{-5}] } :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 = 2 \cdot 3 = (1 + \sqrt{-5}) (1 - \sqrt{-5}). }
It turns out that 2, 3, and the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \pm \sqrt{-5} } are each irreducible. As 1 and -1 are the only numbers dividing 1 in this system, these are two essentially different factorizations of 6 into irreducible numbers.
To see that the above numbers are irreducible, we need a tool, the field norm, which is the function from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [\sqrt{-5}] } to the integers given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N (a + b \sqrt{-5}) = a^2 + 5 b^2.}
Give properties, proof of UF here
Number systems with unique factorization
As mentioned above, unique factorization holds in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [\sqrt{-1}] } , but in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [\sqrt{-5}] } , numbers do not have unique factorizations into irreducible factors. It was the search for a remedy for this that led, first to the invention of ideal numbers by Kummer, and later to the development of ideals and ring theory by Dedekind. Eventually, unique factorization in this setting was salvaged by proving that the principal ideal generated by a number (and actually every ideal) has a unique factorization into prime ideals. For more on this, read about Dedekind domains.
A ring is a mathematical structure similar to the integers, of which the integers, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} [\sqrt{-1}] } , and are all examples. Rings are the appropriate mathematical setting in which to consider the concepts of irreducible factors and unique factorization. Those rings in which unique factorization holds are called unique factorization domains.
![{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a37499ef27234d8a67a65932184280bb17301312)