‫اﻟﻤﺤﺎﺿﺮة اﻟﺜﺎﻣﻨﺔ‬
‫ ﻧﺤ`ﺼﻞ‬R ‫ﻓﻲ اﻟﻤﺤﺎﺿﺮات اﻟﺴﺎﺑﻘﺔ ﻻﺣﻈﻨﺎ اﻧﮫ ﺑﺈﺿﺎﻓﺔ ﺷﺮوط ﻋﻠﻰ ﻋﻤﻠﯿﺔ اﻟﻀﺮب ﻓﻲ اﻟﺤﻠﻘﺔ‬
‫ ﻣﻊ ﻋﻤﻠﯿﺔ اﻟﻀﺮب زﻣﺮة‬R ‫ أﻋﻠﻰ اﻟﺸﺮوط اﻟﻤﻤﻜﻨﺔ ھﻲ أن ﺗﻜﻮن‬. ‫ﻋﻠﻰ أﻧﻮاع ﺧﺎﺻﺔ ﻣﻦ اﻟﺤﻠﻘﺎت‬
‫ أﻣﺎ‬.‫ وﻟﻜﻦ ھﺬا ﻏﯿﺮ ﻣﻤﻜﻦ إﻻ ﻋﻨﺪﻣﺎ ﺗﻜﻮن اﻟﺤﻠﻘﺔ ﺗﺎﻓﮭﺔ )ﺗﺤﺘﻮي ﻋﻠﻰ اﻟﺼﻔﺮ ﻓﻘﻂ( )ﻟﻤﺎذا(؟‬،‫اﺑﺪاﻟﯿﺔ‬
‫ زﻣﺮة اﺑﺪاﻟﯿﺔ )ﻣﻊ ﻋﻤﻠﯿﺔ اﻟﻀﺮب( ﻓﺎن ذﻟ`ﻚ ﻣﻤﻜﻨ`ﺎ وﻓ`ﻲ ھ`ﺬه اﻟﺤﺎﻟ`ﺔ‬R-{0} ‫إذا اﺷﺘﺮطﻨﺎ أن ﺗﻜﻮن‬
.( field) ‫ﻧﺤﺼﻞ ﻋﻠﻰ ﻧﻈﺎم ﺟﺒﺮي أرﻗﻰ ﻣﻦ اﻟﺤﻠﻘﺔ ﯾﺴﻤﻰ اﻟﺤﻘﻞ‬
Definition 1.18 A ring F is said to be field provided that the set F – {0}
is a commutative group under multiplication of F ( the identity of this
group will be written as 1).
:‫ﻣﻼﺣﻈﺔ‬
:‫ﻣﻦ اﻟﺘﻌﺮﯾﻒ أﻋﻼه ﻧﺴﺘﻨﺘﺞ ﻣﻌﻠﻮﻣﺎت اﺿﺎﻓﯿﺔ ﻋﻦ اﻟﺤﻘﻞ ﻧﻠﺨﺼﮭﺎ ﺑﻤﺎ ﯾﻠﻲ‬
.‫ اﻟﺤﻘﻞ ﯾﺤﺘﻮي ﻋﻠﻰ اﻷﻗﻞ ﻋﻨﺼﺮ واﺣﺪ ﻏﯿﺮ ﺻﻔﺮي‬.(‫ ﻋﻤﻠﯿﺔ اﻟﻀﺮب ھﻲ اﺑﺪاﻟﯿﺔ ﻟﺠﻤﯿﻊ ﻋﻨﺎﺻﺮ اﻟﺤﻘﻞ )ﺑﻤﺎ ﻓﯿﮭﺎ اﻟﺼﻔﺮ‬.(‫ ھﻮ ﻋﻨﺼﺮ ﻣﺤﺎﯾﺪ ﻟﺠﻤﯿﻊ ﻋﻨﺎﺻﺮ اﻟﺤﻘﻞ )ﺑﻤﺎ ﻓﯿﮭﺎ اﻟﺼﻔﺮ‬1 ‫ اﻟﻌﻨﺼﺮ اﻟﻤﺤﺎﯾﺪ‬‫ ھﻲ زﻣﺮة ) ﺑﺪون ﻓﺮض اﻻﺑﺪاﻟﯿﺔ( ﻓﺎن اﻟﻨﻈﺎم اﻟﺤﺎﺻﻞ‬F-{0} ‫ ﺣﻠﻘﺔ ﻣﻊ اﻟﺸﺮط أن‬F ‫إذا ﻛﺎﻧﺖ‬
.(skew field) ‫( أو‬division ring) ‫ﯾﺴﻤﻰ‬
Example 1.21
The rational numbers Q and the real numbers R # , with respect to ordinary
addition and multiplication, are the most obvious illustration of fields.
More interesting is the following subset of R # :
F = { a + b 2 | a, bÎ Q}.
Example 1.22
Consider the set C = R # ´ R # of ordered pairs of real numbers, with the
following two operations:
(a, b) + (c, d) = (a+ c, b +d); (a, b)(c, d) = (ac – bd, ad + bc).
In the next an example of a division ring which is not a field will be
discussed .
Example 1.23
Let the set H consist of ordered 4-tuples of real numbers:
H = { (a, b, c, d)| a, b, c, dÎ R # }.
Addition and multiplication on H are defined by the rules
(a, b, c, d) + ( a ¢ , b¢ , c¢ , d ¢ ) = (a + a ¢ , b + b¢ , c + c¢ , d + d ¢ )
(a, b, c, d)( a ¢ , b¢ , c¢ , d ¢ ) = (a a ¢ ̶ b b¢ ̶ c c¢ ̶ d d ¢ ,
a b¢ + b a ¢ + c d ¢ ̶ d̶ c¢ , a c¢ ̶ b d ¢ +c a ¢ +d b¢ , a d ¢ + b c¢ ̶ c b¢ + d a ¢ )
The details will be given in the class.
‫ ( اﻟﺘﻲ ﺗﻢ‬integral domain) ‫اﻟﻤﺒﺮھﻨﺎت اﻟﺘﺎﻟﯿﺔ ﺗﻮﺿﺢ اﻟﻌﻼﻗﺔ ﺑﯿﻦ اﻟﺤﻘﻞ واﻟﺴﺎﺣﺔ اﻟﺘﻜﺎﻣﻠﯿﺔ‬
.‫ وﺳﯿﻌﻄﻰ اﻟﺒﺮھﺎن داﺧﻞ اﻟﺼﻒ‬.‫اﻹﺷﺎرة إﻟﯿﮭﺎ ﻓﻲ ﻣﺤﺎﺿﺮات ﺳﺎﺑﻘﺔ‬
Theorem 1.23 Every field is an integral domain.
Theorem 1.24 Any integral domain with only a finite number of ideals is
a field.
Corollary Any finite integral domain is a field.
‫اﻟﻤﺤﺎﺿﺮة اﻟﺘﺎﺳﻌﺔ‬
‫ واﻻن ﺳ``ﻨﺪرس ﺣﻠﻘ``ﺔ‬،‫ﻓ``ﻲ اﻟﻤﺤﺎﺿ``ﺮة اﻟ `ﺴﺎﺑﻘﺔ درﺳ``ﻨﺎ اﻟﻌﻼﻗ``ﺔ ﺑ``ﯿﻦ اﻟﺤﻘ``ﻞ واﻟ``ﺴﺎﺣﺔ اﻟﺘﻜﺎﻣﻠﯿ``ﺔ‬
‫ واﻟ`ﺴﺆال‬،‫ وﻗﺪ ﻋﻠﻤﻨﺎ ﺳﺎﺑﻘﺎ اﻧﮭﺎ ﺣﻠﻘﺔ اﺑﺪاﻟﯿﺔ ذات ﻣﺤﺎﯾﺪ‬، ( Z n ‫ ) ﻧﻌﻨﻲ‬n ‫اﻷﻋﺪاد اﻟﺼﺤﯿﺤﺔ ﻣﻘﯿﺎس‬
.‫اﻻن ﻣﺘﻰ ﺗﻜﻮن ھﺬه اﻟﺤﻠﻘﺔ ﺣﻘﻼ؟‬
Theorem 1.25 A nonzero element [a] Î Z n is invertible in the ring Z n if
and only if a and n are relatively prime integers.
Proof: in the class.
Corollary. The zero divisors of Z n are precisely the elements of Z n that
are not invertible.
.‫ﯾﻤﻜﻨﻨﺎ ﺟﻤﻊ اﻟﻨﺘﺎﺋﺞ أﻋﻼه ﻓﻲ اﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ‬
Theorem 1.26 The ring Z n of integers modulo n is a field if and only if
n is a prime number. If n is composite, then Z n is not an integral domain,
and the zero divisors of Z n are those nonzero elements [a] for which
gcd(a, n) ¹ 1.
‫( وﻣﻦ ﺧﻼل‬1 ¹ 0 ‫ﻋﻠﻤﻨﺎ ﻣﻦ ﺗﻌﺮﯾﻒ اﻟﺤﻘﻞ اﻧﮫ ﯾﺠﺐ أن ﯾﺤﺘﻮي ﻋﻠﻰ اﻷﻗﻞ ﻋﻨﺼﺮان )ﻻن‬
. Z 2 ‫اﻟﻨﺘﺎﺋﺞ أﻋﻼه ﻧﻠﻤﺲ وﺟﻮد ﺣﻘﻞ ﯾﺤﺘﻮي ﻓﻘﻂ ﻋﻠﻰ اﻟﺤﺪ اﻷدﻧﻰ ﻣﻦ ﻋﺪد اﻟﻌﻨﺎﺻﺮ وھﻮ اﻟﺤﻘﻞ‬
‫ﻣﻦ ﻧﺎﺣﯿﺔ أﺧﺮى ﯾﻮﺟﺪ ﺗﻄﺒﯿﻖ ﻻﻓﺖ ﻟﻠﻨﻈﺮ ﻟﻠﻤﺒﺮھﻨﺔ أﻋﻼه ﻧﻠﺨﺼﮫ ﻓﯿﻤﺎ ﯾﻠﻲ وﻧﺘﺮك اﻟﺘﻔﺼﯿﻞ‬
.‫ﻟﻠﺼﻒ‬
Application. If there exists a homomorphism f: Z → F of the ring Z of
integers onto a field F, then F is necessarily a finite field with prime
number of elements.
‫ﻣﻦ اﺟﻞ ﻋﺮض ﺗﻄﺒﯿﻖ آﺧﺮ ﻣﻔﯿﺪ ﻟﻠﻤﺒﺮھﻨﺔ أﻋﻼه ﻧﺤﺘﺎج أوﻻ إﻟﻰ ﺗﻌﺮﯾﻒ داﻟﺔ ﺣﺴﺎﺑﯿﺔ وھﻲ داﻟﺔ‬
.‫اوﯾﻠﺮ‬
Definition 1.19 The Euler phi-function j is defined as follows: j (1)=1
and, for each integer n> 1, j (n) is the number of invertible elements in
the ring Z n .
Note. By virtue of theorem 1.25, j (n) may also characterized as the
number of positive integers < n that are relatively prime to n. For
instance j (6)=2, j (9)= 6, and j (12)= 4 ; it should be equally clear that
whenever p is a prime number, then j (p)= p – 1.
Lemma If G n is the subset of Z n defined by
G n = { [a] Î Z n | a is relatively prime to n},
then (G n , × n ) forms a finite group of order j (n).
Proof: In the light of the preceding remarks, (G n , × n ) is simply the group
of invertible elements of Z n .
.‫اﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ )واﻟﻤﻌﺮوﻓﺔ ﺑﺎﺳﻢ اﻟﻌﺎﻟﻢ اوﯾﻠﺮ( ﺗﺒﯿﻦ أھﻤﯿﺔ اﺳﺘﺨﺪام اﻟﺠﺒﺮ ﻓﻲ ﻧﻈﺮﯾﺔ اﻷﻋﺪاد‬
Theorem 1.27 ( Euler-Fermat) If n is a positive integer and a is
relatively prime to n, then a j( n ) º 1(mod n).
Proof: The congruence class [a] can be viewed as an element of the
multiplicative group (G n , × n ). Since this group has order j (n), it follows
that [a] j( n ) = [1] or equivalently, a j( n ) º 1(mod n). ( Recall that if G is a
finite group of order k, then x k = 1 for all x in G.)
.‫ ﺗﺘﻀﺢ ﻟﻨﺎ ﻓﻲ اﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ‬،‫ھﻨﺎك ﻋﻼﻗﺔ ﻣﮭﻤﺔ ﺑﯿﻦ اﻟﺤﻘﻞ وﻋﺪد اﻟﻤﺜﺎﻟﯿﺎت‬
Theorem 1.28 Let R be a commutative ring with identity. Then R is a
field if and only if R has no nontrivial ideals.
Proof: In the class.
‫اﻟﻤﺒﺮھﻨﺔ اﻷﺧﯿﺮة ﺗﺆدي إﻟﻰ ﺗﺤﺪﯾﺪ طﺒﯿﻌﺔ أي ﺗﺸﺎﻛﻞ ﺑﯿﻦ ﺣﻘﻠﯿﻦ وھﻮ ﻣﺎ ﺳﻨﺮاه ﻓﻲ اﻟﻤﺒﺮھﻨﺔ اﻟﺘﺎﻟﯿﺔ‬
Theorem 1.29 Let f be a homomorphism from the field F into the field
F¢ . Then either f is the trivial homomorphism or else f is one-to-one.
Proof: Since ker f is an ideal of the field F, either ker f = {0} or else
ker f = F. The first case implies f is one-to-one, and the second means
that f =0.
Corollary. Any homomorphism of a field F onto itself is an
automorphism of F.
Problems 3.
éa 0 ù
1. Let F consist of all matrices in M 2 (Q) of the form ê ú . Prove
ë0 a û
that F is a field.
2. If r is a root of the equation x 2 + x +1=0, show that the set
F= { a +br | a, bÎ Q} forms a field under ordinary addition and
multiplication.
3. a) Assuming that R is a division ring, prove that center R forms a
field.
b) Show that, in a field, every subring with identity is an integral
domain.
4. For any field F, establish that the additive group is not isomorphic
to the multiplicative group. [ Hint: If f : F-{0} → F is an
isomorphism, consider f ((-1) 2 ).]
5. In the field C of complex numbers, define the mapping f : C→
by sending each number to its conjugate; in other words
f ( a+ bi) = a – bi. Verify that f is an automorphism of C.
6. Prove that the product F 1 ´ F 2 of two fields F 1 and F 2 is not a field.
7. Consider the following subset of M 2 (R # );
éa
bù
K={ ê
| a, b Î R # }. Prove that K forms a field that is
ú
ë- b a û
isomorphic to the field C of complex numbers . [ Hint: Define the
éa b ù
ú .]
ë- b a û
mapping f : C→ K by f ( a+ bi) = ê