Virtual Labs

Tools

Aim

Theory

Pretest

Procedure

Simulation

Posttest

References

Contributors

Feedback

Aim

Theory

Pretest

Procedure

Simulation

Posttest

References

Contributors

Feedback

Basics of Finite Fields

Choose difficulty:

Beginner

Intermediate

Advanced

1. Find the multiplicative inverse of 7 in

\mathbb{F}_{11}

?

a: 4 Explanation

Explanation

b: 6 Explanation

Explanation

c: 8 Explanation

Explanation

d: 9 Explanation

Explanation

2. Let

a(x) = 3x^5+4x^3+2x^2+1

and

b(x) = x^3+3x

are belong to

\mathbb{F}_{5}[x]

. Using Division algorithm if it is expressed as

a(x) = q(x) b(x) + r(x)

then

a:

q(x) = x^2+x

and

r(x) = x^2+1

. Explanation

Explanation

b:

q(x) = 3x^2

and

r(x) = 2x^2+1

. Explanation

Explanation

c:

q(x) = 2x^2+x

and

r(x) = x^2+1

. Explanation

Explanation

d: None of the above. Explanation

Explanation

3. Let

f(x) = x^4+x^2+1

belong to

\mathbb{F}_{2}[x]

. The polynomial

f(x)

is

a: irreducible. Explanation

Explanation

b: reducible and

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

. Explanation

Explanation

c: reducible and

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

. Explanation

Explanation

d: None of the above. Explanation

Explanation

4. Consider the field

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

. Let

\alpha

is the root of

(x^3+x+1)

, i.e.,

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

. Then the polynomial representation of

\alpha^5

is

a:

\alpha^2+\alpha

. Explanation

Explanation

b:

\alpha^3+\alpha^2

. Explanation

Explanation

c:

\alpha^2+1

. Explanation

Explanation

d:

\alpha^2+\alpha+1

. Explanation

Explanation

5. Let

f(x) = x^5+x^3+x+1

and

g(x) = x^2+x

are belong to

\mathbb{F}_{2}[x]

. The

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

is equal to

a:

x

. Explanation

Explanation

b:

x+1

. Explanation

Explanation

c:

1

. 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) = 1

and

t(x) = x^2+x

Explanation

Explanation

b:

s(x) = x

and

t(x) = x^2+x

Explanation

Explanation

c:

s(x) = x^2+1

and

t(x) = x^3+x

Explanation

Explanation

d:

s(x) = 1

and

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

Explanation

Explanation

7. Consider the field

\mathbb{F}_{2^2} = \mathbb{F}_2[x]/(x^2+x+1) = \{ 0, 1, \alpha, \alpha^{2} = \alpha+1 \}

. The minimal polynomial of an element

\alpha

is

a:

x+\alpha

. Explanation

Explanation

b:

x^2+1

. Explanation

Explanation

c:

1

. Explanation

Explanation

d:

x^2+x+1

. Explanation

Explanation

Community Links Sakshat Portal Outreach Portal FAQ: Virtual Labs

Contact Us Phone: General Information: 011-26582050 Email: support@vlabs.ac.in

Follow Us

AGPL 3.0 & Creative Commons (CC BY-NC-SA 4.0)