(Solution) 189-235A: Basic Algebra I Assignment 3 Due: Monday, October 21 1. Perform The Euclidean Algorithm To Nd The Gcd Of F (x) = X4 + 3x3 + 16x2 + 33x + 55... | Snapessays.com


(Solution) 189-235A: Basic Algebra I Assignment 3 Due: Monday, October 21 1. Perform the Euclidean algorithm to nd the gcd of f (x) = x4 + 3x3 + 16x2 + 33x + 55...


6. Let d be a fixed integer. Let n=pq ? Z be an integer which is a product of two distinct primes, p and q, and let f ? Z/nZ[x] be a monic polynomial with coefficients in Z/nZ 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/nZ, 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.)189-235A: 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 non-zero element in this ring can be expressed as a power of

 

x

 

.

 

1

 


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