Is the field extension normal?

In summary, the speaker has been asked to determine if certain field extensions are normal. They determine the minimal polynomial for a given element in the extension and check if all of its zeros are also in the extension. If so, the extension is normal. In the example of Q(1+i):Q, the minimal polynomial is x^2-2x+2 and its zeros are 1+i and 1-i, which are both in the extension, making it normal. However, in the example of Q(2^(1/3)):Q, the minimal polynomial has two imaginary roots that are not in the extension, making it not normal.
  • #1
futurebird
272
0
I've been asked to find out if some field extensisons are normal. I want to know if I'm thinking about these in the right way.

For Q(a):Q

I first find the minimal polynomial for a in Q[a]. Then I look at all zeros of that polynomial. If all of the zeros are in Q(a) the extension is normal.


Example:

Q(1+i):Q

1+i = x
-1 = x^2-2x+1

x^2-2x+2 is irreducible over Q and the minimal polynomial of 1+i.

the zereos are: 1+i, 1-i

they are both in Q(1+i) so this is a normal extension.


Correct?
 
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  • #2
*bump*
 
  • #3
Yes, it's normal. Do you know why field extensions might not be normal?
 
  • #4
Q(2^(1/3)):Q is not normal since the minimal polynomial has two imaginary roots that are not in Q(2^(1/3)). Is that the right idea?
 
  • #5
I forgot to say that 2^(1/3) is the real cube root of two.
 
  • #6
Yes, I think so. In your first example, including one root automatically includes the other. In the second it doesn't include the other roots.
 

Related to Is the field extension normal?

1. What does it mean for a field extension to be normal?

A normal field extension is one in which every irreducible polynomial that has one root in the extension field also has all of its other roots in that same extension field.

2. How do you determine if a field extension is normal?

One way to determine if a field extension is normal is by checking if all irreducible polynomials that have one root in the extension field also have all of their other roots in that same extension field.

3. Can a non-normal field extension still be useful in mathematics or science?

Yes, a non-normal field extension can still be useful in certain mathematical or scientific applications. For example, non-normal field extensions may be used in cryptography or coding theory.

4. What are some examples of normal and non-normal field extensions?

Examples of normal field extensions include the complex numbers over the real numbers and the algebraic closure of a finite field. Non-normal field extensions include the field of real numbers over the rational numbers and the field of algebraic numbers over the rational numbers.

5. How does the concept of normality relate to other properties of field extensions?

The concept of normality is closely related to other properties of field extensions such as separability and algebraic closure. A normal field extension is always separable, meaning every irreducible polynomial over the base field has distinct roots in the extension field. Additionally, a normal field extension is always algebraically closed, meaning every irreducible polynomial over the base field has a root in the extension field. However, the converse is not always true, as there are separable and algebraically closed field extensions that are not normal.

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