Prove an extension is not normal

In summary, a normal extension in mathematics is a Galois extension that preserves certain algebraic properties. To prove that an extension is not normal, one can check if it satisfies the necessary conditions such as being Galois and preserving roots of polynomials. Some examples of non-normal extensions include the extension of rational numbers by the cube root of 2, the square root of 3, and the golden ratio. It is important to prove that an extension is not normal in order to gain a better understanding of different types of extensions and their properties, which can lead to new mathematical discoveries and applications. Practical applications for normal and non-normal extensions can be found in fields such as cryptography, coding theory, engineering, and physics.
  • #1
PsychonautQQ
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Homework Statement


let b be a square root of 1+i, show that Q(b):Q is not a normal extension. Also, what is the Galois group of the extension?

Homework Equations

The Attempt at a Solution


so b = +/- (1+i)^(1/2), and it's minimal polynomial is x^4+4 which has roots -(2)^1/2 and 2^(1/2) that are not in Q(b) and therefore the extension is not normal. In a proper splitting field, x^4+4 splits into (x-(1+i))(x+(1+i))(x-2^(1/2))(x+2^(1/2)), the Galois group would have a map that permutes the roots of the first two factors and a map that would permute the roots of the second two factors, therefore it would be isomorphic to Z_2 x Z_2.

Is this correct?
 
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  • #2
PsychonautQQ said:
it's minimal polynomial is x^4+4
That doesn't look right to me.

We have ##b^2=1+i## so that ##b^2-1=i## so that ##(b^2-1)^2=-1##. Hence a monic 4th degree polynomial with root ##b## is ##(x^2-1)^2+1=x^4-2x^2+2##. Since the minimal polynomial is unique, I don't think your polynomial can also be minimal.

The structure of your argument is otherwise OK. There will be four roots of the minimal polynomial, two of which are in ##\mathbb Q(b)## and two of which are not.
 
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  • #3
On the second part of the question, you appear to have found the Galois group of the extension that includes the other two roots of that minimal polynomial, which is not what was asked. Nor will the Galois group just be ##\mathbb Z_2##. This part of this question has many similarities to that earlier one you asked about an extension based on ##c##, a primitive fourteenth root of 1. Here the extension is based on ##b##, a primitive 16th root of ##2^8##.

Remember how each fixing automorphism in that other example mapped ##c## to ##c^k## where ##k\geq 1## and ##k\leq## (did you ever end up working out what the upper limit was?).

Well here the same thing will apply. Each fixing automorphism will map ##b## to a positive integer power of ##b##, up to a certain upper limit.

If you can solve that other problem, including finding that upper limit, you can solve this one.
 
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FAQ: Prove an extension is not normal

What does it mean for an extension to be "normal"?

In mathematics, an extension is considered normal if it is a Galois extension, meaning it is the splitting field of a separable polynomial over its base field. This means that the extension preserves certain algebraic properties, such as the roots of the polynomial being preserved in the extension field.

How can I prove that an extension is not normal?

To prove that an extension is not normal, you can use the definition of a normal extension and check if it satisfies all the necessary conditions. This may involve checking if the extension is Galois, if it is a splitting field, and if it preserves certain algebraic properties. If the extension fails to meet any of these conditions, then it can be proven to not be normal.

What are some examples of non-normal extensions?

One example of a non-normal extension is the extension of the rational numbers by the cube root of 2. This extension is not normal because it does not preserve the roots of certain polynomials, such as x^3 - 2. Other examples of non-normal extensions include the extension of the rational numbers by the square root of 3 and the extension of the rational numbers by the golden ratio.

Why is it important to prove that an extension is not normal?

Proving that an extension is not normal can be important in fields such as algebra and number theory. It helps us understand the properties of different types of extensions and can lead to the discovery of new mathematical concepts and theories. Additionally, knowing when an extension is not normal can also help us determine when certain mathematical methods or techniques may not be applicable.

Are there any practical applications for normal and non-normal extensions?

Yes, there are practical applications for normal and non-normal extensions in fields such as cryptography and coding theory. Normal extensions are used in cryptographic algorithms to ensure security and prevent attacks, while non-normal extensions can be used in coding theory to create error-correcting codes. Additionally, understanding the properties of extensions can also help in solving practical problems in fields such as engineering and physics.

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