Let E be an algebraic over F, F is perfect. Show that E is perfect

  • Thread starter barbiemathgurl
  • Start date
In summary, the conversation discusses proving that an algebraic extension E over a perfect field F is also perfect. The concept of "perfect" is clarified as all extensions being separable. The conversation then moves on to giving a step-by-step approach to proving the statement, involving showing that the irreducible polynomial of an element in E over E has zeros of multiplicity 1 and demonstrating the relation between irreducible polynomials over a base field and a larger field. The notation irr<a,E> is explained as the monic irreducible polynomial over E with "a" as a zero.
  • #1
barbiemathgurl
12
0
let E be an algebraic over F where F is perfect. Show that E is perfect. :uhh:
 
Physics news on Phys.org
  • #2
Well, what have you tried? Our purpose here is to help you work through a problem, not to be an answer book!
 
  • #3
what doies perfect mean? all algebvraic extensions are separable? sounds trivial from the definitions doesn't it? have you thought about it?

if so, and it still eludes you, think abiout the relation between the irreducible polynomial for a given element over a base field as opposed to over a larger field.
 
Last edited:
  • #4
@mathwonk: Yes "perfect" means all extensions are seperable.

@barbie: Do the following
1)Let [tex]K/E<\infty[/tex].
2)Let [tex]\alpha \in E[/tex]
3)Show [tex]\mbox{irr} \left< \alpha, E \right>[/tex] has zeros of multiplicity 1 (Hint: Show that [tex]\mbox{irr}\left< \alpha, E \right> | \mbox{irr} \left< \alpha, F \right>[/tex]*



*)If K algebraic over E algebraic over F then K algebraic over F.
 
  • #5
Just out of interest: What does the notation irr<a,E> mean?

I though maybe it was a notation for the minimal polynomial of a over E, but as a was in E that doesn't make much sense your 3 wouldn't make much sense.
 
  • #6
Palindrom said:
Just out of interest: What does the notation irr<a,E> mean?

It is the monic irreducible polynomial such that contains "a" as a zero. This notation is non-standard, but saves space which I have seen in one book.
 

1. What does it mean for a field to be "perfect"?

A field F is considered perfect if every irreducible polynomial over F has distinct roots in its algebraic closure. This means that every polynomial equation over F has a solution in the algebraic closure of F.

2. How do you prove that E is perfect if F is perfect?

To prove that E is perfect if F is perfect, we must show that every irreducible polynomial over E has distinct roots in its algebraic closure. Since E is an extension of F, every irreducible polynomial over E can be factored into irreducible polynomials over F. And since F is perfect, each of these factors has distinct roots in its algebraic closure. Therefore, E is also perfect.

3. Can you give an example of an algebraic field that is not perfect?

Yes, an example of an algebraic field that is not perfect is the field of rational functions over a finite field. In this field, the polynomial x^p - t has no roots, where p is the characteristic of the field and t is an element that is not a pth power. This shows that not all irreducible polynomials have distinct roots in the algebraic closure, making it not perfect.

4. How does the concept of perfection relate to algebraic closure?

Algebraic closure and perfection are closely related concepts. A field F is perfect if every irreducible polynomial over F has distinct roots in its algebraic closure. This means that in order to prove that a field is perfect, we must consider its algebraic closure. On the other hand, if a field is not perfect, then there exist irreducible polynomials that do not have distinct roots in its algebraic closure.

5. Why is it important for a field to be perfect?

The concept of perfection is important in many areas of mathematics, including algebraic geometry and number theory. For example, in algebraic geometry, the concept of perfection is used to study varieties and their points. In number theory, perfection is important in understanding the behavior of prime numbers in certain fields. Furthermore, the concept of perfection allows us to classify fields in terms of their properties, making it a useful tool in exploring the structure of fields.

Similar threads

  • Linear and Abstract Algebra
Replies
19
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
10
Views
197
Replies
5
Views
452
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
19
Views
1K
Replies
9
Views
910
Replies
2
Views
908
Back
Top