Let p be a prime such that p mod 4 = 3. How many elements oforder 4 are in GF(pn)*?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

Let p be a prime such that p mod 4 = 3. How many elements of
order 4 are in GF(pn)*?

Expert Solution
Step 1

Given: A group G= GF(pn) and the relation is p3 mod4.

To Find: Number of elements of order 4.

Step 2

Let G be a group GF(pn).

the given relation is p3 mod4....................(i)

We know that GF(pn) is a multiplicative group which is isomorphic to pn-1.

pn-1 has an element of order 4 if 4|pn-1.

this implies there exists k, such that,

pn-1=4k...................(ii)

from (i),

pn3n mod4.

Now from (ii),

4k+13n mod44k(3n-1) mod4.

Now since 4k is always divisible by 4,

3n-1=0,4,8,12,....

thus the possibilities of are,

n=2,4,6,8,...

 n is even.

Hence both relations (i) and (ii) are possible when n is even.

steps

Step by step

Solved in 3 steps

Blurred answer
Knowledge Booster
Groups
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,