Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Usc the q notiont ving firld construction tu find extension K of Q such that k contains a 2 f(x)= x³ + x² +2 d root of the

Usc the q notiont ving firld construction tu find extension K of Q such that k contains a 2 f(x)= x³ + x² +2 d root of the

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
Usc the q notiont ving construction to find field extension K of Q such that k k S contains a 2 f(x)= fo) x3 tr tả cho a root of this
Transcribed Image Text:Usc the q notiont ving construction to find field extension K of Q such that k k S contains a 2 f(x)= fo) x3 tr tả cho a root of this
Expert Solution
Step 1: Main Steps

To find a field extension K of Q such that K contains a root of  f(x)=x3+x2+2, we can use the quotient ring construction as follows:

First,

 we need to find an irreducible polynomial g(x) in Q[x] that divides f(x). One possible choice is g(x)=x2+x+1, which is irreducible over Q by the rational root test or by Eisenstein’s criterion with p = 2.


Second,

we form the quotient ring,Q[x]/(g(x)) where (g(x)) is the ideal generated by g(x) in Q[x]. This quotient ring is a field, since        g(x) is irreducible and hence (g(x)) is a maximal ideal.


steps

Step by step

Solved in 3 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