How to prove normality of a field extension with Q(\sqrt{2+\sqrt{2}}):Q?

In summary, the problem is to show that the field extension Q(\sqrt{2+\sqrt{2}}):Q is normal. This can be done by showing that the element \sqrt{2-\sqrt{2}} is also an element of the field, which is necessary for the minimum polynomial to split over the field. Several approaches can be taken, such as solving for \sqrt{2-\sqrt{2}} as a function of \beta, using common manipulations to find its conjugate, or using the Galois group of the splitting field of the minimal polynomial. It is recommended to try all three approaches to find a solution.
  • #36
Thanks, I'll take a look at it.
 
<h2> How do I determine if Q(√2+√2):Q is a field extension?</h2><p>To determine if Q(√2+√2):Q is a field extension, we must first check if Q(√2+√2) is a field. This can be done by verifying that every element in Q(√2+√2) has an inverse in Q(√2+√2). We also need to check if Q is a subfield of Q(√2+√2).</p><h2> What is the degree of Q(√2+√2):Q?</h2><p>The degree of Q(√2+√2):Q is 4. This is because the minimal polynomial of √2+√2 over Q is x^4 - 4x^2 + 2, which is a quartic polynomial.</p><h2> How can I show that Q(√2+√2):Q is a normal extension?</h2><p>To prove that Q(√2+√2):Q is a normal extension, we need to show that it is a splitting field for some polynomial over Q. In this case, we can show that Q(√2+√2) is a splitting field for the polynomial x^4 - 4x^2 + 2 over Q.</p><h2> Can I use the Galois correspondence to prove the normality of Q(√2+√2):Q?</h2><p>Yes, the Galois correspondence can be used to prove the normality of Q(√2+√2):Q. We can show that the Galois group of Q(√2+√2):Q over Q is isomorphic to the Klein four-group, which is a normal subgroup of the symmetric group S4. This implies that Q(√2+√2):Q is a normal extension.</p><h2> Are there any other methods to prove the normality of Q(√2+√2):Q?</h2><p>Yes, there are other methods to prove the normality of Q(√2+√2):Q. One approach is to use the fact that Q(√2+√2) is a quadratic extension of Q(√2), which is a normal extension. Another method is to use the fact that Q(√2+√2) is a Galois extension, and therefore, it must be a normal extension.</p>

FAQ: How to prove normality of a field extension with Q(\sqrt{2+\sqrt{2}}):Q?

How do I determine if Q(√2+√2):Q is a field extension?

To determine if Q(√2+√2):Q is a field extension, we must first check if Q(√2+√2) is a field. This can be done by verifying that every element in Q(√2+√2) has an inverse in Q(√2+√2). We also need to check if Q is a subfield of Q(√2+√2).

What is the degree of Q(√2+√2):Q?

The degree of Q(√2+√2):Q is 4. This is because the minimal polynomial of √2+√2 over Q is x^4 - 4x^2 + 2, which is a quartic polynomial.

How can I show that Q(√2+√2):Q is a normal extension?

To prove that Q(√2+√2):Q is a normal extension, we need to show that it is a splitting field for some polynomial over Q. In this case, we can show that Q(√2+√2) is a splitting field for the polynomial x^4 - 4x^2 + 2 over Q.

Can I use the Galois correspondence to prove the normality of Q(√2+√2):Q?

Yes, the Galois correspondence can be used to prove the normality of Q(√2+√2):Q. We can show that the Galois group of Q(√2+√2):Q over Q is isomorphic to the Klein four-group, which is a normal subgroup of the symmetric group S4. This implies that Q(√2+√2):Q is a normal extension.

Are there any other methods to prove the normality of Q(√2+√2):Q?

Yes, there are other methods to prove the normality of Q(√2+√2):Q. One approach is to use the fact that Q(√2+√2) is a quadratic extension of Q(√2), which is a normal extension. Another method is to use the fact that Q(√2+√2) is a Galois extension, and therefore, it must be a normal extension.

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