Need help understanding splitting fields

In summary, the conversation discusses finding a splitting field for the polynomial x^4 - 6x^2 - 7 over the rational numbers. The polynomial factors to (x^2 - 7)*(x^2+1), with roots of 7^(1/2) and i. The extension field E is determined to be Q(i)(7^(1/2)), but there is confusion about the textbook jumping straight to Q(i)(2^(1/2)). It is questioned if the square root of 7 is an element of the simple extension of the rational numbers with the square root of two. A mathamaverick is asked for clarification and it is suggested that the result may be correct if not looking for a
  • #1
PsychonautQQ
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My textbook is going through an example on splitting fields. It asked to find a splitting field for x^4 - 6x^2 - 7 over the rational numbers. This polynomial factors to (x^2 - 7)*(x^2+1) which has roots of 7^(1/2) and i. So i figured the extension field E we are looking for is Q(i)(7^(1/2)), but my textbook jumps straight to Q(i)(2^(1/2).

is the squareroot of 7 an element of the simple extension of the rational numbers with the square root of two? I can't imagine it being (yet i can't imagine many things that are..).

Any mathamaverick want to shed some light on my situation?
 
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  • #2
Doesn't it follow, if ##2^{1/2}## is in a field extension of the rationals that ##(7/2)( 2^{1/2}) ## is also on the field, e.g., by closure under multiplication? Your result is correct if you re not looking for a minimal extension.
 
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  • #3
Okay, what am i missing? How does (7/2)*(2^(1/2)) show that the finite extensions of 7^(1/2) reduces to 2^(1/2)?
 
  • #4
Ah, sorry, I misread, let me think again.
 
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  • #5
It looks to me like a misprint. No, the extension fields containing p^{1/2} and q^{1/2}, for p and q prime, are NOT the same.
 
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Related to Need help understanding splitting fields

1. What is a splitting field?

A splitting field is a field extension of a given field that contains all the roots of a given polynomial. It is the smallest field extension that contains all the roots of the polynomial.

2. Why is it important to understand splitting fields?

Understanding splitting fields is important because it allows us to find the roots of polynomials over a given field. It also helps us to understand the structure of a given field and its extensions.

3. How do you find the splitting field of a polynomial?

To find the splitting field of a polynomial, we first factor the polynomial into irreducible factors. Then, we adjoin the roots of each irreducible factor to the given field. The resulting field is the splitting field.

4. Can a polynomial have multiple splitting fields?

No, a polynomial can only have one splitting field. This is because the splitting field is defined as the smallest field extension that contains all the roots of the polynomial. Any other field extension containing these roots would be a superset of the splitting field.

5. How are splitting fields related to Galois theory?

Splitting fields are closely related to Galois theory, which is the study of field extensions and their corresponding groups of automorphisms. The Galois group of a polynomial is a subgroup of the symmetric group of its roots, and the structure of the splitting field is determined by the Galois group.

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