8.Let p be a prime and let F denote the field Z/pZ with p elements. Let g(x) be a polynomial in F[x]. show that gcd (x^(p)x, g(x)) is a polynomial whose degree is equal to the number of distinct roots in F.
9. Use the result of question 8 to show that the polynomial x^2+1 has no roots in Z/pZ when p is a prime of the form 3+4m.189235A: Basic Algebra I
Assignment 3
Due: Monday, October 21
1.
Perform the Euclidean algorithm to Fnd the gcd of
f
(
x
) =
x
4
+ 3
x
3
+
16
x
2
+ 33
x
+ 55 and
g
(
x
) =
x
3
+
x
2

x

10 in the polynomial ring
Q
[
x
].
Write this greatest common divisor as a linear combination of
f
(
x
) and
g
(
x
)
with coe±cients in
Q
[
x
].
2.
Same question as 1, with
f
(
x
) =
x
6
+
x
4
+
x
+ 1 and
g
(
x
) =
x
6
+
x
5
+
x
4
+
x
3
+
x
2
+
x
+ 1 in
Z
/
2
Z
[
x
].
3. List all the irreducible polynomials of degree 4 in
Z
/
2
Z
[
x
].
4.
If
p
is an odd prime of the form 1 + 4
m
, use Wilson’s Theorem to show
that
a
= (2
m
)! is a root in
Z
/p
Z
of the polynomial
x
2
+ 1 in
Z
/p
Z
[
x
].
5.
In class, we showed that a polynomial of degree
d
with coe±cients in a
Feld
F
has at most
d
roots. Show that this statement ceases to be true when
F
is replaced by an arbitrary ring, such as the ring
Z
/n
Z
of residue classes
modulo
n
with
n
a composite integer.
6.
Let
d
be a Fxed integer.
Let
n
=
pq
?
Z
be an integer which is a
product of two distinct primes,
p
and
q
, and let
f
?
Z
/n
Z
[
x
] be a monic
polynomial with coe±cients in
Z
/n
Z
of degree
d
.
Give a “best possible”
general upper bound (as a function of
d
) for the number of distinct roots
that such a polynomial could have over
Z
/n
Z
, and show with an example
that your estimate is indeed best possible.
(I.e., describe a judicious choice
of
f
having the maximal number of distinct roots.)
7.
Write down the powers of
x
in the ring
Z
/
2
Z
[
x
]
/
(
x
3
+
x
+ 1) and show
that every nonzero element in this ring can be expressed as a power of
x
.
1
This question was answered on: May 23, 2022
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