Examples of normal and non-normal extensions of Q?

  • Thread starter ae1709
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  • #1
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Hi, I'm really struggling to find examples (with proofs) of the following:

1) For each n>2 give an example of a non-normal extension of Q of degree n.

2) Give examples of normal extensions of Q of degrees 3,4 and 5.

3) Show that for any positive integer n, there exists a normal extension of Q of degree n.

Any help would be much appreciated!
 

Answers and Replies

  • #2
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So an extension [itex]\mathbb{Q}\subseteq K[/itex] is normal if for all [itex]\alpha\in K[/itex] the minimal polynomial of [itex]\alpha[/itex] splits in K. Or equivalently if K is the splitting field of a polynomial in [itex]\mathbb{Q}[/itex].

So, can you adjoin a number [itex]\alpha[/itex] to [itex]\mathbb{Q}[/itex] such that the minimal polynomial doesn't split?? This answers (a).
 
  • #3
Deveno
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for 2) remember that the algebraic closure of Q is not a subfield of R, so you need to look for some complex numbers that make this happen. i suggest looking on the unit circle, perhaps?
 
  • #4
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So an extension [itex]\mathbb{Q}\subseteq K[/itex] is normal if for all [itex]\alpha\in K[/itex] the minimal polynomial of [itex]\alpha[/itex] splits in K. Or equivalently if K is the splitting field of a polynomial in [itex]\mathbb{Q}[/itex].

So, can you adjoin a number [itex]\alpha[/itex] to [itex]\mathbb{Q}[/itex] such that the minimal polynomial doesn't split?? This answers (a).
But adjoining one number will surely not give you an extension of degree n as required in the question?
 
  • #5
22,089
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But adjoining one number will surely not give you an extension of degree n as required in the question?
It might, for example, adjoining [itex]\sqrt[3]{2}[/itex] gives you a nonnormal extension of degree 3.
 

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