Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
If p = 1 mod 4, then we can still define F,[i] as a ring, but it is not a field. Illustrate this in a specific case by finding a non-zero element in F[i] that is not invertible. Hint: in any ring, a zero divisor is an element x such that there is another non-zero element y with xy = 0. A zero divisor is never invertible – indeed, if x is is invertible then x xy -1 = x0 y = 0. xY = 0 Thus it suffices to find a non-zero zero divisor in F5|i].

If p = 1 mod 4, then we can still define F,[i] as a ring, but it is not a field. Illustrate this in a specific case by finding a non-zero element in F[i] that is not invertible. Hint: in any ring, a zero divisor is an element x such that there is another non-zero element y with xy = 0. A zero divisor is never invertible – indeed, if x is is invertible then x xy -1 = x0 y = 0. xY = 0 Thus it suffices to find a non-zero zero divisor in F5|i].

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

I require help with this problem from Number theory

If p = 1 mod 4, then we can still define Fp[i] as a ring, but it is not a field. Illustrate this in a specific case by finding a non-zero element in F;[i] that is not invertible. Hint: in any ring, a zero divisor is an element x such that there is another non-zero element y with xy = 0. A zero divisor is never invertible indeed, if x is is invertible then x-lxy = x='0 = y = 0. xy = 0 Thus it suffices to find a non-zero zero divisor in F5[i].
Transcribed Image Text:If p = 1 mod 4, then we can still define Fp[i] as a ring, but it is not a field. Illustrate this in a specific case by finding a non-zero element in F;[i] that is not invertible. Hint: in any ring, a zero divisor is an element x such that there is another non-zero element y with xy = 0. A zero divisor is never invertible indeed, if x is is invertible then x-lxy = x='0 = y = 0. xy = 0 Thus it suffices to find a non-zero zero divisor in F5[i].
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
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,
SEE MORE TEXTBOOKS