1+2x + x 2 ∈ Z5 [x]

```WORKING FOR EXAM QUESTIONS FOR ALGEBRA 2 VI·SA 2
1. Let f (x) = x4 + 3x3 + 2x2 + 2 and g(x) = 1 + 2x + x2 2 Z5 [x] :
a. Find ( f + g) (x)
b. Find( f g) (x) polynomial.
c. Find the roots of f (x)
d. Find the roots of g (x)
e. Find q (x) ; r (x) 2 Z5 [x] such that f (x) = q (x) g(x)+r (x) ; where either
r (x) = 0 or 0
deg (r (x)) < deg (g(x))
2. In Z8 [x] ; prove the followings.
a. Find the all units of Z8 [x] :
b.Show that 1 + 2x is a unit in the ring Z8 [x] :
c.Show that 4x2 + 2x2 + 4 is a zero divisor
d.Show that 2x is nilpotent
3. For the following statements, write the proof if the statement is true;
otherwise, give a counterexample.
i: If a polynomial ring R [x] has zero divisor, so does R:
ii: If R is a …eld, then R [x] is a …eld.
7
iii:In Z7 [x] ; x + 1 = x7 + 1:
iv: Z is a Euclidean Domain
v: Q is a Euclidean Domain
v{: Z [x] is a Euclidean Domain
v{{: Q [x] is a Euclidean Domain
v{{{: Every Euclidean Domain is Principal ideal domain
x{: In integral domain, irreducible element is prime element
x: Z is a principal ideal domain
x{: Z [i] is a Euclidean domain
x{{: 13 is an irreducible element in Z [i]
x{{{:In Z12 ; every prime element is an irreducible element
p
4. Find all units of the integral domain Z i 3
1
5. In Z7 [x] ; prove the followings.
i: Find the all units of Z7 [x] :
ii:Find all the associates of 2 + x2 in Z7 [x] :
6. Find all the associates of followings
i: Find all the associates of 3 2ipin Z [i] p
ii: Find all the associates of 1 + i 5 in Z i 5
iii.Find all the associates of 6 in Z10
iv.Find all the associates of 2 + xpin Z3 [x] p
v: Find all the associates of 7 + 2
3 in Z
3 :
v{: Find all the associates of 2 + x 3x2 Z [x] :
7. Show that 4 and 6 are associates in Z10
8. Let n be a a square free integer(an integer di¤erent from 0 and 1,which
is not divisible by the square of any integer.) Let
p
p
Z n = a + b n; a; b 2 Z
p
We know that Z [ n] is an integral domain. De…ne a function by
p
N : Z n n f0g ! Z+ [ f0g
p
p
p
a + b n 7! a + b n a b n = a2 nb2
p
i: Let x 2 Z [ n] :Prove that N (x) = 0 () x = 0 p
ii:Prove that Np(xy) = N (x) N (y) for all x; y 2 Z [ n]
p
iii: Let x 2 Z [ n] :Prove that N (x) = 1 () x is a unit in Z [ n]
p
Uygulamada benzer soruyu N (x) = p,(primeinteger) () xis an irreducible in Z [ n] ¸seklinde yapm¬¸st¬k.
2
9. Show that 2 i; 1 + i and 11 are irreducible elements inpZ [i] :
Hint : N (x) = p,(primeinteger)() xis an irreducible in Z [ n]
10.
In the domain Z [i] ; prove the following
i) Find the gcd (2 7i; 2 + 11i)
ii) Find x and y in Z [i] such that gcd (2
7i; 2 + 11i) = x (2
7i)+y (2 + 11i) :
p
p
11.Show that in the integral domain Z i 5 ; 2 + i 5 is an irreducible
element, but not a prime element.
12. In Z12 ;
a. Is 3 a prime element?
b. Is 3 an irreducible element?
p
13.Show that 3 is not prime element in Z i 5 :
14. .a.Is the polynomial x3 + x2 + 1 irreducible in Z2 [x]?
b:Is the polynomial x2 + 1 irreducible in Z2 [x]?
c:Is the polynomial x3 + x2 + 1 irreducible in Z3 [x]?
d:.Is the polynomial x2 + 1 irreducible in Z3 [x]?
15. In Z [i], show that 3 is a prime element, but 5 is not a prime element.
1
BAZI TAVS·
IYELER
Ders notuna çok iyi çal¬¸
s¬p öncelikle dersle ilgili tan¬m ve kavramlar¬
iyice ö¼
grenin.
Baz¬önemli teoremlerin ispat¬n¬bilin.
Hem tan¬mlarda hem teoremlerdeki Ayk¬r¬ örnekleri muhakkak
bilin.
Uygulamada ve derste yap¬lan örnekleri çözüp iyi anlay¬n.
Verilen ödevleri çözün iyi kavray¬n
En son olarakta çal¬¸
sma sorular¬n¬yap¬n.
Lütfen son gece çal¬¸
smay¬n, yeti¸
stiremezsiniz en az¬ndan 2 veya 3
gün öncesinden ba¸
slay¬n.
S¬navda BA¸
SARILAR...
3
4
```
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