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Aim

Theory

Pretest

Procedure

Simulation

Posttest

References

Contributors

Feedback

Basics of Finite Fields

Choose difficulty:

Beginner

Intermediate

Advanced

1. Which of the following can't be a field?

a:

\mathbb{F}_{23}

Explanation

Explanation

b:

\mathbb{F}_{16}

Explanation

Explanation

c:

\mathbb{F}_{27}

Explanation

Explanation

d:

\mathbb{F}_{21}

Explanation

Explanation

2. Consider the field

\mathbb{F}_{9}

. If

gcd(6,4) = 6.x+4.y

then

a:

x=2

and

y=7

. Explanation

Explanation

b:

x=1

and

y=8

. Explanation

Explanation

c:

x=3

and

y=4

. Explanation

Explanation

d:

x=2

and

y=6

. Explanation

Explanation

3. Consider the field

\mathbb{F}_{2^4} = \mathbb{F}_2[x]/(x^4+x^3+1)

. Let

\alpha

is the root of

(x^4+x^3+1)

, i.e.,

\alpha^4+\alpha^{3}+1=0

. Then the polynomial representation of

\alpha^7

is

a:

\alpha^3+\alpha

. Explanation

Explanation

b:

\alpha^3+\alpha^2+\alpha+1

. Explanation

Explanation

c:

\alpha^4+\alpha^2+1

. Explanation

Explanation

d:

\alpha^2+\alpha+1

. Explanation

Explanation

4. Consider the field

\mathbb{F}_{2^4} = \mathbb{F}_2[x]/(x^3+x+1)

. Let

\alpha

is the root of

(x^4+x^3+1)

, i.e.,

\alpha^{3}+\alpha+1=0

. The multiplicative inverse of

\alpha^7

is

a:

\alpha^9

. Explanation

Explanation

b:

\alpha^8

. Explanation

Explanation

c:

\alpha^7

. Explanation

Explanation

d:

\alpha^4

. Explanation

Explanation

5. Let

f(x) = 2x^5+x^4+2x^3+x^2+x

and

g(x) = x^2+2x+1

are belong to

\mathbb{F}_{3}[x]

. Then

gcd(f(x),g(x))

is equal to

a:

2x+1

. Explanation

Explanation

b:

x+2

. Explanation

Explanation

c:

2x+2

. Explanation

Explanation

d: None of the above. Explanation

Explanation

6. For the above problem find the

s(x)

and

t(x)

such that

gcd(f(x),g(x)) = s(x)f(x)+t(x)g(x)

?

a:

s(x) = 2

and

t(x) = 2x^2+x

Explanation

Explanation

b:

s(x) = 1

and

t(x) = x^3+2

Explanation

Explanation

c:

s(x) = 2x^2+1

and

t(x) = x^3+2x

Explanation

Explanation

d:

s(x) = x

and

t(x) = 2x^3+x^2

Explanation

Explanation

7. Consider the field

\mathbb{F}_{2^4} = \mathbb{F}_2[x]/(x^4+x^3+1)

. The minimal polynomial of an element

\alpha^5

is

a:

x^3+x

. Explanation

Explanation

b:

x^2+x+1

. Explanation

Explanation

c:

x^2+1

. Explanation

Explanation

d: None of the above. Explanation

Explanation

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