Questions for the exam
April 27, 2010
1
Theoretical questions
1.
Formulate binomial theorem. Right the formula for (
x
+
y
)
n
and formula for
the binomial coefficients.
2.
For two positive integer numbers
n, m
define
gcd
(
n, m
) and prove that it
exists. You can not use prime number factorization, you can use well ordering
principle.
3.
For two polynomials
n
(
x
)
, m
(
x
) define
gcd
(
n
(
x
)
, m
(
x
)) and prove that it
exists.
4.
Prove that if integers numbers
a
and
c
are relatively prime and
c
divides
ab
then
c
divides
b
.
5.
Prove that if polynomials
a
(
x
) and
c
(
x
) are relatively prime and
c
(
x
) divides
a
(
x
)
b
(
x
) then
c
(
x
) divides
b
(
x
).
6.
Prove that there are infinitely many prime numbers.
7.
Prove that if a product of several numbers is divisible by a prime number
p
then one of them is divisible by
p
.
8.
Prove that if a product of several polynomials is divisible by an irreducible
polynomial
p
(
x
) then one of them is divisible by
p
(
x
).
9.
a)Prove that any positive integer number could be written as a product of
prime numbers
n
=
p
1
p
2
· · ·
p
k
b) Prove that the factorization is unique in the sense that if also
n
=
q
1
q
2
· · ·
q
l
then
l
=
k
and we can renumber the
q
i
, so that
q
i
=
p
i
.
10.
What polynomials are irreducible over
R
? You don’t need to prove it, only
describe them so it was easy to check if polynomial is irreducible or not.
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 Spring '08
 MULLEN
 Math, Binomial Theorem, Binomial, Prime number

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