## Transcribed Text

1. Problem
Which of the following are groups under addition, multiplication, Rings and fields? If
not, why not (explicitly state the properties they do not satisfy)
Group
Group under
Ring with
Field
under
multiplication
identity
addition
(After removing
the additive
identity)
Natural Numbers (1,2,3,
Under ordinary addition and
multiplication
Whole Numbers (0,1,2,3,.. )
Under ordinary addition and
multiplication
Integers
Under ordinary addition and
nultiplication
Rational Numbers Q
Under ordinary addition and
multiplication
Real Numbers
Under ordinary addition and
aultiplication
Complex Numbers
Under complex number addition
and multiplication
Z.
Under Mod addition and
multiplication
Z2
Under Mod addition and
multiplication
2. Problem
Which of the following polynomials defined over the field of real numbers are
irreducible?
f(x) x+1
g(x) =x-1
h(x=x2+1
p(x)=x2-1
q(x) + :
r(x)
3. Problem
Multiply the following polynomials defined over set of real numbers:
f(x) x2 and g(x) =x+ 1.
4. Problem
Using polynomial long division, divide f(x) by g(x) defined over the set of real
numbers:
f(x) x+ x2 and g(x) x + 1.
Find g(x) andr(x) such that f(x) = q(x)g(x) r(x)
5. Problem
Which of the following polynomials defined over the field of GF(2) are irreducible?
f(x) 1
g(x) =x-1
h(x) =x2+1
p(x) x2-1
q(x) =x++1
r(x) =x³
6. Problem
Multiply the following polynomials defined over GF(2):
f(x) x+ +x2+ and g(x) = = x +1.
7. Problem
Using polynomial long division. divide f(x) by g(x) defined over GF(2)
f(x) = x4 x2+ and g(x) + 1.
Find q(x) and r(x) such that f(x) = q(x)g(x) r(x)
8. Problem
In this problem all polynomials are over GF(2):
f(x) X2 + and g(x) =x + 1
Using extended Euclidean algorithm find s(x) and t(x) such that:
god(f(x),g(x)) s(x)f(x) t(x)g(x)
Hint: the procedure is identical to one we used for two numbers a and b where we
discovered and y Here integers a and are replaced by polynomials f(x) and g(x) and
the linear coefficients : and are replaced by polynomials that you have to find s(x) and
t(x). Use long division.

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