Remark 2: Note that there can be several fields of order p". However, they will all be isomorphic, not equal. For example, you can check that x² +1 and x²-x-1 are both irreducible over F3. Hence K= F₂[x] B₂(x)/(x² + 1) ' L=F₂[x] and (x²-x-1) are both fields having 9 elements. Hence both are how do we get this isomorphic to F,. But KL, since, for example, x² +1=0 in K, but x² +1=x+2+0 in L.
Remark 2: Note that there can be several fields of order p". However, they will all be isomorphic, not equal. For example, you can check that x² +1 and x²-x-1 are both irreducible over F3. Hence K= F₂[x] B₂(x)/(x² + 1) ' L=F₂[x] and (x²-x-1) are both fields having 9 elements. Hence both are how do we get this isomorphic to F,. But KL, since, for example, x² +1=0 in K, but x² +1=x+2+0 in L.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Remark 2: Note that there can be several fields of order p". However, they
will all be isomorphic, not equal. For example, you can check that x² +1 and
_F,[x]/
B₂(x)/(x² + 1)
and
x²-x-1 are both irreducible over F. Hence K =
L= F₂[x]/
5(x ) (x²-x-1) are both fields having 9 elements. Hence both are
how do we get this
2
isomorphic to F. But KL, since, for example, x² +1=0 in K, but
x²+1=x+2+Ō in L.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15b7304-cc73-4505-92c3-23aa2fda4f71%2F4aa00b10-9186-4a93-b721-3a67ce343ab7%2Fzhqelmx_processed.png&w=3840&q=75)
Transcribed Image Text:Remark 2: Note that there can be several fields of order p". However, they
will all be isomorphic, not equal. For example, you can check that x² +1 and
_F,[x]/
B₂(x)/(x² + 1)
and
x²-x-1 are both irreducible over F. Hence K =
L= F₂[x]/
5(x ) (x²-x-1) are both fields having 9 elements. Hence both are
how do we get this
2
isomorphic to F. But KL, since, for example, x² +1=0 in K, but
x²+1=x+2+Ō in L.
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