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jmjlt88
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Simple Field Extensions... Misleading Question?
This is from Pinter's A Book of Abstract Algebra. The section of exercises is called "Simple Extensions."
[page 278]
3. a+c is a root of p(x) iff a is a root of p(x+c) ...
I was thinking about the question in terms of field extensions. I attempted to use the fact that F(a)=F(a+c), which I proved in a preceding exercise. That is, the field generated by F and a is equal to the field generated by F and (a+c). But, I simply noticed that ...
If a+c is a root of p(x), then p(a+c)=0. Now, considering the polynomial p(x+c), if we plug in a we get p(a+c), which we know equals 0. [The converse is similar.]
Seemed way too easy, and thus I am very nervous that I missed something, especially since the section focused on properties of simple field extension.
Am I correct? Thanks for the help! =)
This is from Pinter's A Book of Abstract Algebra. The section of exercises is called "Simple Extensions."
[page 278]
3. a+c is a root of p(x) iff a is a root of p(x+c) ...
I was thinking about the question in terms of field extensions. I attempted to use the fact that F(a)=F(a+c), which I proved in a preceding exercise. That is, the field generated by F and a is equal to the field generated by F and (a+c). But, I simply noticed that ...
If a+c is a root of p(x), then p(a+c)=0. Now, considering the polynomial p(x+c), if we plug in a we get p(a+c), which we know equals 0. [The converse is similar.]
Seemed way too easy, and thus I am very nervous that I missed something, especially since the section focused on properties of simple field extension.
Am I correct? Thanks for the help! =)