Simple Field Extensions Misleading Question?

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In summary, the conversation is about a question from Pinter's A Book of Abstract Algebra, specifically from the section called "Simple Extensions." The question is about whether a+c being a root of a polynomial p(x) is equivalent to a being a root of p(x+c). The conversation includes a proposed answer to the question and a discussion about its placement in the book. Ultimately, it is determined that the question is meant as a hint towards a later exercise.
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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! =)
 
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  • #2


Yeah, your proof seems correct. It's a pretty weird question, let me check pinter.
 
  • #3


I guess that Pinter includes that exercise as some kind of "hint" towards exercise 4.
 
  • #4


micromass, thank you so much! I am glad you feel its a little weird too. It seems misplaced. The previous chapter was entitled "Substitution in Polynomials." This question, in my opinion, would fit nicely there. One of the great things about Pinter is that the heading of each exercise set gets the reader thinking in a certain direction. But, when an odd question shows up, the novice starts to feel a little doubtful when the answer required nothing from the section. =)
 
  • #5


Oh, there we go! Just took a look at exercise 4. Good ol' Pinter, never disappoints! :)
 

1. What are simple field extensions?

Simple field extensions are a type of field extension in algebraic field theory. They are extensions of a field by adjoining a single element that satisfies a simple algebraic equation. This allows for the creation of a larger field from a smaller one.

2. How are simple field extensions different from other field extensions?

Simple field extensions are distinguished by the fact that they are generated by a single element, as opposed to multiple elements in other types of field extensions. This makes them easier to study and understand.

3. Can simple field extensions be used in real-world applications?

Yes, simple field extensions have many applications in mathematics and other fields such as physics and engineering. They are used to solve various problems involving algebraic equations and to construct larger fields for specific purposes.

4. Are simple field extensions always simple?

No, simple field extensions are not always simple. In some cases, they can be further extended to create even larger fields, called composite field extensions. However, the process of creating a composite field extension involves multiple steps and is not as straightforward as creating a simple field extension.

5. How can simple field extensions be visualized?

Simple field extensions can be visualized using geometric objects called field extension diagrams. These diagrams show the relationship between the elements of the original field and the extended field, and can help in understanding the structure of the field extension.

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