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| {{Distinguish|Fundamental theorem of algebra}}
| | Hello! <br>My name is Gemma and I'm a 25 years old boy from Italy.<br><br>Here is my web-site: [http://Shop.uneedadv.com/index.php/User:GenaDamico fifa 15 Coin generator] |
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| In [[number theory]], the '''fundamental theorem of arithmetic''', also called the '''unique factorization theorem''' or the '''unique-prime-factorization theorem''', states that every [[integer]] greater than 1 either is prime itself or is the product of [[prime number]]s, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.<ref>{{harvtxt|Long|1972|p=44}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=53}}</ref><ref>Hardy & Wright, Thm 2</ref> For example,
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| : <math>
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| 1200
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| = 2^4 \times 3^1 \times 5^2
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| = 3 \times 2\times 2\times 2\times 2 \times 5 \times 5
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| = 5 \times 2\times 3\times 2\times 5 \times 2 \times 2
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| =\cdots\text { etc.}
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| \!
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| </math>
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| The theorem is stating two things: first, that 1200 ''can'' be represented as a product of primes, and second, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.
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| The requirement that the factors be prime is necessary: factorizations containing [[composite number]]s may not be unique (e.g. 12 = 2 × 6 = 3 × 4).
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| ==History==
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| Book VII, propositions 30 and 32 of [[Euclid]]'s [[Euclid's Elements|Elements]] is essentially the statement and proof of the fundamental theorem. Article 16 of [[Carl Friedrich Gauss|Gauss]]' ''[[Disquisitiones Arithmeticae]]'' is an early modern statement and proof employing [[modular arithmetic]].
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| ==Applications==
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| ===Canonical representation of a positive integer=== <!-- Redirect [[Canonical form]] links here -->
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| Every positive integer ''n'' > 1 can be represented '''in exactly one way''' as a product of prime powers:
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| :<math>
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| n
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| = p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}
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| = \prod_{i=1}^{k}p_i^{\alpha_i}
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| </math>
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| where {{nowrap begin}}''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>k</sub>{{nowrap end}} are primes and the α<sub>''i''</sub> are positive integers.
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| This representation is called the '''canonical representation'''<ref>{{harvtxt|Long|1972|p=45}}</ref> of ''n'', or the '''standard form'''<ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=55}}</ref><ref>Hardy & Wright § 1.2</ref> of ''n''.
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| :For example 999 = 3<sup>3</sup>×37, 1000 = 2<sup>3</sup>×5<sup>3</sup>, 1001 = 7×11×13
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| Note that factors ''p''<sup>0</sup> = 1 may be inserted without changing the value of ''n'' (e.g. 1000 = 2<sup>3</sup>×3<sup>0</sup>×5<sup>3</sup>).<br>In fact, any positive integer can be uniquely represented as an [[infinite product]] taken over all the positive prime numbers,
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| :<math>
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| n=2^{n_2}3^{n_3}5^{n_5}7^{n_7}\cdots=\prod p_i^{n_{p_i}}.
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| </math>
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| where a finite number of the ''n''<sub>''i''</sub> are positive integers, and the rest are zero. Allowing negative exponents provides a canonical form for positive [[rational number]]s.
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| ===Arithmetic operations===
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| This representation is convenient for expressions like these for the product, [[greatest common divisor|gcd]], and [[least common multiple|lcm]]:
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| :<math>
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| a\cdot b
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| =2^{a_2+b_2}\,3^{a_3+b_3}\,5^{a_5+b_5}\,7^{a_7+b_7}\cdots
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| =\prod p_i^{a_{p_i}+b_{p_i}},
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| </math>
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| :<math>
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| \gcd(a,b)
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| =2^{\min(a_2,b_2)}\,3^{\min(a_3,b_3)}\,5^{\min(a_5,b_5)}\,7^{\min(a_7,b_7)}\cdots
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| =\prod p_i^{\min(a_{p_i},b_{p_i})},
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| </math>
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| :<math>
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| \operatorname{lcm}(a,b)
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| =2^{\max(a_2,b_2)}\,3^{\max(a_3,b_3)}\,5^{\max(a_5,b_5)}\,7^{\max(a_7,b_7)}\cdots
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| =\prod p_i^{\max(a_{p_i},b_{p_i})}.
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| </math>
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| While expressions like these are of great theoretical importance their practical use is limited by our ability to [[Integer factorization|factor]] numbers.
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| ===Arithmetical functions===
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| {{Main|Arithmetic function}}
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| Many arithmetical functions are defined using the canonical representation. In particular, the values of [[additive function|additive]] and [[multiplicative function|multiplicative]] functions are determined by their values on the powers of prime numbers.
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| ==Proof==
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| The proof uses [[Euclid's lemma]] (''Elements'' VII, 30): if a prime ''p'' [[Divisor|divides]] the product of two [[natural number]]s ''a'' and ''b'', then ''p'' divides ''a'' or ''p'' divides ''b'' (or both). The article has proofs of the lemma.
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| ===Existence===
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| By induction: assume it is true for all numbers between 1 and ''n''. If ''n'' is prime, there is nothing more to prove. Otherwise, there are integers ''a'' and ''b'', where ''n'' = ''ab'' and {{nowrap begin}}1 < ''a'' ≤ ''b'' < ''n''.{{nowrap end}}
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| By the induction hypothesis,
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| {{nowrap begin}}''a'' = ''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''n''</sub>{{nowrap end}}
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| and
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| {{nowrap begin}}''b'' = ''q''<sub>1</sub>''q''<sub>2</sub>...''q''<sub>''m''</sub>{{nowrap end}} are products of primes. But then
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| {{nowrap begin}}''n'' = ''ab'' = ''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''n''</sub>''q''<sub>1</sub>''q''<sub>2</sub>...''q''<sub>''m''</sub>{{nowrap end}} is the product of primes. In the base case, 2 is a trivial product of primes.
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| ===Uniqueness===
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| Assume that ''s'' > 1 is the product of prime numbers in two different ways:
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| :<math>
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| \begin{align}
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| s
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| &=p_1 p_2 \cdots p_m \\
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| &=q_1 q_2 \cdots q_n.
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| \end{align}
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| </math>
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| We must show ''m'' = ''n'' and that the ''q''<sub>''j''</sub> are a rearrangement of the ''p''<sub>''i''</sub>.
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| By [[Euclid's lemma]], ''p''<sub>1</sub> must divide one of the ''q''<sub>''j''</sub>; relabeling the ''q''<sub>''j''</sub> if necessary, say that ''p''<sub>1</sub> divides ''q''<sub>1</sub>. But ''q''<sub>1</sub> is prime, so its only divisors are itself and 1. Therefore, ''p''<sub>1</sub> = ''q''<sub>1</sub>, so that
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| :<math>
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| \begin{align}
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| \frac{s}{p_1}
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| &=p_2 \cdots p_m \\
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| &=q_2 \cdots q_n.
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| \end{align}
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| </math>
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| Reasoning the same way, ''p''<sub>2</sub> must equal one of the remaining ''q''<sub>''j''</sub>. Relabeling again if necessary, say ''p''<sub>2</sub> = ''q''<sub>2</sub>. Then
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| :<math>
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| \begin{align}
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| \frac{s}{p_1 p_2}
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| &=p_3 \cdots p_m \\
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| &=q_3 \cdots q_n.
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| \end{align}
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| </math>
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| This can be done for all ''m'' of the ''p''<sub>''i''</sub>, showing that ''m'' ≤ ''n''. If there were any ''q''<sub>''j''</sub> left over we would have
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| :<math>
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| \begin{align}
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| \frac{s}{p_1 p_2 \cdots p_m}
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| &=1 \\
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| &=q_k \cdots q_n,
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| \end{align}
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| </math>
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| which is impossible, since the product of numbers greater than 1 cannot equal 1. Therefore ''m'' = ''n'' and every ''q''<sub>''j''</sub> is a ''p''<sub>''i''</sub>.
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| ===Elementary proof of uniqueness===
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| The fundamental theorem of arithmetic can also be proved without using Euclid's lemma, as follows:
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| Assume that ''s'' > 1 is the smallest positive integer which is the product of prime numbers in two different ways. If ''s'' were prime then it would factor uniquely as itself, so there must be at least two primes in each factorization of ''s'':
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| :<math>
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| \begin{align}
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| s
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| &=p_1 p_2 \cdots p_m \\
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| &=q_1 q_2 \cdots q_n.
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| \end{align}
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| </math>
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| If any ''p''<sub>''i''</sub> = ''q''<sub>''j''</sub> then, by cancellation, ''s''/''p''<sub>''i''</sub> = ''s''/''q''<sub>''j''</sub> would be a positive integer greater than 1 with two distinct factorizations. But ''s''/''p''<sub>''i''</sub> is smaller than ''s'', meaning ''s'' would not actually be the smallest such integer. Therefore every ''p''<sub>''i''</sub> must be distinct from every ''q''<sub>''j''</sub>.
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| Without loss of generality, take ''p''<sub>1</sub> < ''q''<sub>1</sub> (if this is not already the case, switch the ''p'' and ''q'' designations.) Consider
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| :<math>t = (q_1 - p_1)(q_2 \cdots q_n),</math>
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| and note that 1 < ''q''<sub>2</sub> ≤ ''t'' < ''s''. Therefore ''t'' must have a unique prime factorization. By rearrangement we see,
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| :<math>
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| \begin{align}
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| t
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| &= q_1(q_2 \cdots q_n) - p_1(q_2 \cdots q_n) \\
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| &= s - p_1(q_2 \cdots q_n) \\
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| &= p_1((p_2 \cdots p_m) - (q_2 \cdots q_n)).
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| \end{align}
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| </math>
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| Here ''u'' = ((''p''<sub>2</sub> ... ''p''<sub>''m''</sub>) - (''q''<sub>2</sub> ... ''q''<sub>''n''</sub>)) is positive, for if it were negative or zero then so would be its product with ''p''<sub>''1''</sub>, but that product equals ''t'' which is positive. So ''u'' is either 1 or factors into primes. In either case, ''t'' = ''p''<sub>1</sub>''u'' yields a prime factorization of ''t'', which we know to be unique, so ''p''<sub>1</sub> appears in the prime factorization of ''t''. | |
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| If (''q''<sub>1</sub> - ''p''<sub>1</sub>) equaled 1 then the prime factorization of ''t'' would be all ''q'''s, which would preclude ''p''<sub>1</sub> from appearing. Thus (''q''<sub>1</sub> - ''p''<sub>1</sub>) is not 1, but is positive, so it factors into primes: (''q''<sub>1</sub> - ''p''<sub>1</sub>) = (''r''<sub>1</sub> ... ''r''<sub>''h''</sub>). This yields a prime factorization of
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| :<math>t = (r_1 \cdots r_h)(q_2 \cdots q_n),</math> | |
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| which we know is unique. Now, ''p''<sub>1</sub> appears in the prime factorization of ''t'', and it is not equal to any ''q'', so it must be one of the ''r'''s. That means ''p''<sub>1</sub> is a factor of (''q''<sub>1</sub> - ''p''<sub>1</sub>), so there exists a positive integer ''k'' such that ''p''<sub>1</sub>''k'' = (''q''<sub>1</sub> - ''p''<sub>1</sub>), and therefore
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| :<math>p_1(k+1) = q_1.</math>
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| But that means ''q''<sub>1</sub> has a proper factorization, so it is not a prime number. This contradiction shows that ''s'' does not actually have two different prime factorizations. As a result, there is no smallest positive integer with multiple prime factorizations, hence all positive integers greater than 1 factor uniquely into primes.
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| ==Generalizations==
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| The first generalization of the theorem is found in Gauss's second monograph (1832) on [[biquadratic reciprocity]]. This paper introduced what is now called the [[ring theory|ring]] of [[Gaussian integer]]s, the set of all [[complex number]]s ''a'' + ''bi'' where ''a'' and ''b'' are integers. It is now denoted by <math>\mathbb{Z}[i].</math> He showed that this ring has the four units ±1 and ±''i'', that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes.<ref>Gauss, BQ, §§ 31–34</ref>
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| Similarly, in 1844 while working on [[cubic reciprocity]], [[Gotthold Eisenstein|Eisenstein]] introduced the ring <math>\mathbb{Z}[\omega]</math>, where <math>\omega=\frac{-1+\sqrt{-3}}{2}, </math> <math>\omega^3=1 </math> is a cube [[root of unity]]. This is the ring of [[Eisenstein integer]]s, and he proved it has the six units <math>\pm 1, \pm\omega, \pm\omega^2</math> and that it has unique factorization.
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| However, it was also discovered that unique factorization does not always hold. An example is given by <math>\mathbb{Z}[\sqrt{-5}]</math>. In this ring one has<ref>Hardy & Wright, § 14.6</ref>
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| :<math>
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| 6=
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| 2\times 3=
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| (1+\sqrt{-5})\times(1-\sqrt{-5}).
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| </math>
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| Examples like this caused the notion of "prime" to be modified. In <math>\mathbb{Z}[\sqrt{-5}]</math> it can be proven that if any of the factors above can be represented as a product, e.g. 2 = ''ab'', then one of ''a'' or ''b'' must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; e.g.
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| 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6. In [[algebraic number theory]] 2 is called '''irreducible''' (only divisible by itself or a unit) but not '''prime''' (if it divides a product it must divide one of the factors). Using these definitions it can be proven that in any ring a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers <math>\mathbb{Z}</math> every irreducible is prime". This is also true in <math>\mathbb{Z}[i]</math> and <math>\mathbb{Z}[\omega],</math> but not in <math>\mathbb{Z}[\sqrt{-5}].</math>
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| The rings where every irreducible is prime are called [[unique factorization domain]]s. As the name indicates, the fundamental theorem of arithmetic is true in them. Important examples are [[Euclidean domain]]s and [[principal ideal domain]]s.
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| In 1843 [[Ernst Kummer|Kummer]] introduced the concept of [[ideal number]], which was developed further by [[Richard Dedekind|Dedekind]] (1876) into the modern theory of [[Ideal (ring theory)|ideals]], special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called [[Dedekind domain]]s.
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| ==See also==
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| * [[Riemann zeta function#Euler product formula|Euler product formula]]
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| * [[Integer factorization]]
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| * [[Noetherian ring]]
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| * [[Prime signature]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
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| * {{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Clarke | first2 = Arthur A. (translator into English)
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| | title = Disquisitiones Arithemeticae (Second, corrected edition)
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | location = New York
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| | year = 1986
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| | isbn = 0387962549}}
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| * {{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Maser | first2 = H. (translator into German)
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| | title = Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
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| | publisher = Chelsea
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| | location = New York
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| | year = 1965
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| | isbn = 0-8284-0191-8}}
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| The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § ''n''". Footnotes referencing the ''Disquisitiones Arithmeticae'' are of the form "Gauss, DA, Art. ''n''".
| |
| * {{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | title = Theoria residuorum biquadraticorum, Commentatio prima
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| | publisher = Comment. Soc. regiae sci, Göttingen 6
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| | location = Göttingen
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| | year = 1828}}
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| * {{citation
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| | last1 = Gauss | first1 = Carl Friedrich
| |
| | title = Theoria residuorum biquadraticorum, Commentatio secunda
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| | publisher = Comment. Soc. regiae sci, Göttingen 7
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| | location = Göttingen
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| | year = 1832}}
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| These are in Gauss's ''Werke'', Vol II, pp. 65–92 and 93–148; German translations are pp. 511–533 and 534–586 of the German edition of the ''Disquisitiones''.
| |
| * {{Citation
| |
| | last=Baker
| |
| | first=Alan
| |
| | author-link=
| |
| | year=1984
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| | title=A Concise Introduction to the Theory of Numbers
| |
| | place=Cambridge, UK
| |
| | publisher=Cambridge University Press
| |
| | isbn=978-0-521-28654-1
| |
| }}
| |
| * {{Citation
| |
| | last1=Hardy
| |
| | first1=G. H.
| |
| | author1-link=G. H. Hardy
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| | last2=Wright
| |
| | first2=E. M.
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| | author2-link=E. M. Wright
| |
| | year=1979
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| | title=An Introduction to the Theory of Numbers
| |
| | edition=fifth
| |
| | place=USA
| |
| | publisher=Oxford University Press
| |
| | isbn=978-0-19-853171-5
| |
| }}
| |
| * {{Citation
| |
| | author1 = A. Kornilowicz
| |
| | author2 = P. Rudnicki
| |
| | year = 2004
| |
| | title = Fundamental theorem of arithmetic
| |
| | journal = [[Formalized Mathematics]]
| |
| | volume = 12
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| | issue = 2
| |
| | pages = 179–185
| |
| }}
| |
| * {{citation
| |
| | first1 = Calvin T.
| |
| | last1 = Long
| |
| | year = 1972
| |
| | title = Elementary Introduction to Number Theory
| |
| | edition = 2nd | publisher = [[D. C. Heath and Company]]
| |
| | location = Lexington
| |
| | lccn = 77-171950
| |
| }}.
| |
| * {{citation
| |
| | first1 = Anthony J.
| |
| | last1 = Pettofrezzo
| |
| | first2 = Donald R.
| |
| | last2 = Byrkit
| |
| | year = 1970
| |
| | title = Elements of Number Theory
| |
| | publisher = [[Prentice Hall]]
| |
| | location = Englewood Cliffs
| |
| | lccn = 77-81766
| |
| }}.
| |
| * {{citation
| |
| | last1 = Riesel | first1 = Hans
| |
| | title = Prime Numbers and Computer Methods for Factorization (second edition)
| |
| | publisher = Birkhäuser
| |
| | location = Boston
| |
| | year = 1994
| |
| | isbn = 0-8176-3743-5}}
| |
| * {{mathworld|urlname=AbnormalNumber|title=Abnormal number}}
| |
| * {{mathworld|urlname=FundamentalTheoremofArithmetic|title=Fundamental Theorem of Arithmetic}}
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| | |
| == External links ==
| |
| * [http://www.cut-the-knot.org/blue/gcd_fta.shtml GCD and the Fundamental Theorem of Arithmetic] at [[cut-the-knot]].
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| * [http://planetmath.org/fundamentaltheoremofarithmeticproofofthe PlanetMath: Proof of fundamental theorem of arithmetic]
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| * [http://fermatslasttheorem.blogspot.com/2005/06/unique-factorization.html Fermat's Last Theorem Blog: Unique Factorization], a blog that covers the history of [[Fermat's Last Theorem]] from [[Diophantus of Alexandria]] to the proof by [[Andrew Wiles]].
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| * [http://demonstrations.wolfram.com/FundamentalTheoremOfArithmetic/ "Fundamental Theorem of Arithmetic"] by Hector Zenil, [[Wolfram Demonstrations Project]], 2007.
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| * {{cite web|last=Grime|first=James|title=1 and Prime Numbers|url=http://www.numberphile.com/videos/1notprime.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| {{Fundamental theorems}}
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| {{Divisor classes}}
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| [[Category:Theorems about prime numbers]]
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| [[Category:Articles containing proofs]]
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| [[Category:Fundamental theorems|Arithmetic]]
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| [[de:Primfaktorzerlegung#Fundamentalsatz der Arithmetik]]
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