Basic properties of the integers Chapter 1. 1.1 Divisibility and primality Theorem 1.1 For all a, b,...
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Transcript of Basic properties of the integers Chapter 1. 1.1 Divisibility and primality Theorem 1.1 For all a, b,...
![Page 1: Basic properties of the integers Chapter 1. 1.1 Divisibility and primality Theorem 1.1 For all a, b, c Z, (i) a|a, 1|a, and a|0; (ii) 0|a if and only.](https://reader038.fdocuments.net/reader038/viewer/2022110206/56649cf45503460f949c2c93/html5/thumbnails/1.jpg)
Basic properties of the integers
Chapter 1
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1.1 Divisibility and primalityTheorem 1.1 For all a, b, c Z,(i) a|a, 1|a, and a|0;(ii) 0|a if and only if a = 0;(iii) a|b if and only if a|b if and only if a|b;(iv) a|b and a|c implies a|(b + c);(v) a|b and b|c implies a|c.
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Theorem 1.2 For all a, b Z, we have a|b and b|a if and only
if a = b. In particular, for every a Z, we have a|1 if an
d only if a = 1.
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Primes and composites If n > 1 and no other positive integers besid
es 1 and n divide n, then we say n is prime. If n > 1 but n is not prime, then we say that n
is composite. Evidently, n is composite if and only if n = a
b for some integers a, b with 1 < a < n and 1 < b < n.
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Fundamental theorem of arithmetic Theorem 1.3 Every non-zero integer n can be expressed as
where p1, . . . , pr are distinct primes and e1, . . . , er are positive integers.
Moreover, this expression is unique, up to a reordering of the primes.
rer
ee pppn 2121
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Proof of Theorem 1.3 Existence part: By induction on n, every positive integer
n can be expressed as a product (possibly empty) of primes.
If n = 1, the statement is true, as n is the product of zero primes.
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Proof of Theorem 1.3 Now let n > 1, and assume that every
positive integer smaller than n can be expressed as a product of primes.
If n is a prime, then the statement is true, as n is the product of one prime.
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Proof of Theorem 1.3 Assume that n is composite, so that there exi
st a, b Z with 1 < a < n, 1 < b < n, and n = ab.
By the induction hypothesis, both a and b can be expressed as a product of primes, and so the same holds for n.
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Proof of Theorem 1.3 Uniqueness part: the hard part
Need other theorems.
Theorem 1.4 (Division with remainder property). Let a, b Z with b > 0. Then there exist unique q, r Z such that a = bq + r and 0 r < b.
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Proof of Theorem 1.4 Existence of r and q. Consider the set S of non-negative integers of
the form a - bt with t Z. This set is clearly non-empty; indeed, if a 0,
set t := 0, and if a < 0, set t := a.
Since every non-empty set of non-negative integers contains a minimum, we define r to be the smallest element of S.
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Proof of Theorem 1.4 By definition, r is of the form r = a - bq for som
e q Z, and r 0.
We must have r < b, since otherwise, r - b would be an element of S smaller than r, contradicting the minimality of r; indeed, if r b, then we would have 0 r b = a - b(q + 1).
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Proof of Theorem 1.4 Uniqueness of r and q. Suppose that a = bq + r and a = bq’ + r’, wh
ere 0 r< b and 0 r’ < b.
Then subtracting these two equations and rearranging terms, we obtain r’ - r = b(q – q’).
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Proof of Theorem 1.4 Thus, r’ - r is a multiple of b. However, 0 r < b and 0 r’< b implies |
r’-r|< b. Therefore, the only possibility is r
’- r = 0 = b(q – q’) and b 0 implies q – q’ = 0.
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The mod operator Let a, b Z with b > 0. If q and r are the unique integers from Theo
rem 1.4 that satisfy a = bq + r and 0 r< b, we define a mod b := r;
(a mod b) = a - ba/b.
Theorem 1.3 is still unsolved and will be dealt later!
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1.2 Ideals and greatest common divisors A non-empty set I Z is an ideal if and onl
y if for all a, b I and all z Z, we have a + b I and az I.
Every ideal I contains 0. If an ideal I contains an integer a, it also con
tains -a, since a = -a (-1) I.
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If an ideal I contains a and b, it also contains a - b.
{0} and Z are ideals. An ideal I = Z if and only if 1 I
For a Z, define aZ := {az : z Z} aZ is an ideal
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aZ is called the ideal generated by a
An ideal of the form aZ for some a Z is called a principal ideal.
Suppose I1 and I2 are ideals. I1 + I2 := {a1 + a2 : a1 I1, a2 I2} is also an ideal.
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Example 1.1. 3Z = {…,-9,-6,-3,0,3,6,9,…}.
Example 1.2. Consider the ideal 3Z + 5Z. This ideal contains 3 2 + 5 (-1) = 1. Thus, 3Z + 5Z = Z.
Example 1.3. Consider the ideal 4Z + 6Z. This ideal contains 4 (-1) + 6 1 = 2. Thus, 4Z + 6Z = 2Z.
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Every ideal is a principal ideal Theorem 1.6. Let I be an ideal of Z. Then there exists a uni
que non-negative integer d such that I = dZ.
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Proof of Theorem 1.6 Existence part: assume that I {0}. Since I contains non-zero integers, it must c
ontain positive integers, since if a I then so is -a.
Let d be the smallest positive integer in I.
Claim: I = dZ.
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dZ I since d I.
I dZ. Let a be any element in I It suffices to show that d | a. a = dq + r, where 0 r < d. r = a – dq is also an element of I, and by the
minimality of the choice of d, we must have r = 0. Thus, d | a.
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Proof of Theorem 1.6 Uniqueness part: If dZ = eZ for some non-negative integer e, t
hen d | e and e | d, d=e or –e d,e>0 implies d=e.
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Greatest common divisors For a, b Z, we call d Z a common
divisor of a and b if d | a and d | b.
Call such a d a greatest common divisor of a and b if d is non-negative and all other common divisors of a and b divide d
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Greatest common divisors Theorem 1.7. For all a, b Z, there exists a u
nique greatest common divisor d of a and b, and moreover, aZ + bZ = dZ
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Proof of Theorem 1.7 We apply Theorem 1.6 to the ideal I := aZ + b
Z. Let d Z with I = dZ, as in Theorem 1.6
Claim: d is a greatest common divisor of a and b
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Proof of Theorem 1.7 a I = dZ, we see that d | a; similarly, d | b d is a common divisor of a and b.
Since d I = aZ + bZ, there exist s, t Z such that as + bt = d.
Suppose a = a’d’ and b = b’d’ for some a’, b’, d’ Z
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Proof of Theorem 1.7 as + bt = d implies that d’(a’s+b’t) = d,
which says that d’| d. d is a greatest common divisor of a and b.
For uniqueness, if e is a greatest common divisor of a and b, then d | e and e | d, and hence d = e.
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For a, b Z, we write gcd(a, b) for the greatest common divisor of a and b.
We say that a, b Z are relatively prime if gcd(a, b) = 1
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Theorem 1.8. Let a, b, r Z and let d := gcd(a, b). Then there exist s, t Z such that as + bt = r if and only if d | r.
In particular, a and b are relatively prime if and only if there exist integers s and t such that as + bt = 1.
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gcd(a, b) may be characterized as the largest positive integer that divides both a and b, and as the smallest positive integer that can be expressed as as + bt for integers s and t.
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Theorem 1.9. Let a, b, c Z such that c | ab and gcd(a, c) = 1. Then c | b.
Suppose that c | ab and gcd(a, c) = 1. Then since gcd(a, c) = 1, by Theorem 1.8 we
have as + ct = 1 for some s, t Z. Multiplying this equation by b, we obtain a
bs + cbt = b.
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Proof of Theorem 1.9 c divides ab by hypothesis c clearly divides cbt Hence that c divides b.
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Theorem 1.10. Let p be prime, and let a, b Z. Then p | ab implies that p | a or p | b.
Assume that p | ab. If p | a, we are done, so assume that p doesn’t divide a.
gcd(a, p) = 1, and so by Theorem 1.9, we have p | b.
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Finishing the proof of Theorem 1.3 Suppose that p1, . . . , pr are primes (not nec
essarily distinct), and q1, . . . , qs are primes (also not necessarily distinct), such that p1…pr = q1…qs
Claim: (p1,…,pr) is just a reordering of (q1,…,qs)
Prove it by induction on r.
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Finishing the proof of Theorem 1.3 If r = 0, we must have s = 0 and we are done. Now suppose r > 0, and that the statement
holds for r - 1. Since r > 0, we clearly must have s > 0. p1…pr = q1…qs p1 | q1…qs By Theorem 1.10, p1 | qj for some j Since qj is prime, we must have p1 = qj. The theorem follows from induction.
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1.3 Some consequences of unique factorization Theorem 1.11 There are infinitely many pri
mes.
By way of contradiction, suppose that there were only finitely many primes.
Call them p1, . . . , pk.
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Proof of Theorem 1.11
Set M := p1p2...pk and N := M + 1. There must be at least one prime p such tha
t p | N. p cannot equal any of the pi’’s. Otherwise, p | M=N-1 and p | N which is imp
ossible.