Splitting Fields and Separable Polynomials ...

  • #1
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I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ...

I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the definitions differently in examples ... I need help to understand why these things appear different and what the significance and implications of the differences are ...


D&F define separable polynomial ... and give an example as follows:


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Bland defines separable polynomials as follows ... and also gives an example ...



?temp_hash=fa2f76956f2b960bd41ad6e66b9bb839.png

?temp_hash=fa2f76956f2b960bd41ad6e66b9bb839.png





My questions are as follows:


Question 1


Now ... for Bland, to qualify to be a separable polynomial, a polynomial must be irreducible ... and then it must have no non-distinct roots ...

For D&F any polynomial that has no multiple roots is separable ...


Is this difference in definitions significant?

Which is the more usual definition?



Question 2


In D&F in Example 1 we are given a polynomial ##f(x) = x^2 - 2## as an example of a separable polynomial ...

... and ... D&F also as us to consider ##(x^2 - 2)^n## for ##n \ge 2## as inseparable as it has repeated or multiple roots ##\pm \sqrt{2}## ...

... a particular case would be ##(x^2 - 2)^2## and a similar analysis would mean ##(x^2 + 2)^2## would also be inseparable ...


BUT ...

Bland analyses the polynomial ##f(x) = (x^2 + 2)^2 ( x^2 - 3)## and comes to the conclusion that ##f## is separable ... when I think that D&Fs analysis would have found the polynomial to be inseparable ...

Can someone explain and reconcile the differences in D&F and Bland's approaches and solutions ... ...




Question 3


In D&F Example 1 we read ...

" ... ... The polynomial ##x^2 - 2## is separable over ##\mathbb{Q}## ... ... "


I am curious and somewhat puzzled and perplexed about how the term "over" applies to a separable polynomial ... both D&F and Bland define separability in terms of distinct or non-multiple roots ... they do not really define separability OVER something ...

Can someone explain how "over" comes into the definition and how a polynomial can be separable over one field but not separable over another ... ?




Hope that someone can help ...

Peter
 

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Answers and Replies

  • #2
mathwonk
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i am not an expert, but i think that the key definition is of a separable irreducible polynomial, where both definitions agree. i.e. i think the main point is to define a separable extension, which only uses the concept of a separable minimal (hence irreducible) polynomial. i may be wrong, but i would bet on this version.

I have just looked it up in zariski samuel where they define separable to be distinct roots in the irreducible case, and in thje general case to be that all irreducible factors are separable. they are definitely authoritative to me at least. this seems to be bland's definition. but note that it does not give any different version of the theory of separable field extensions. i hope. check that out.
 
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  • #3
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i am not an expert, but i think that the key definition is of a separable irreducible polynomial, where both definitions agree. i.e. i think the main point is to define a separable extension, which only uses the concept of a separable minimal (hence irreducible) polynomial. i may be wrong, but i would bet on this version.

I have just looked it up in zariski samuel where they define separable to be distinct roots in the irreducible case, and in thje general case to be that all irreducible factors are separable. they are definitely authoritative to me at least. this seems to be bland's definition. but note that it does not give any different version of the theory of separable field extensions. i hope. check that out.


Thanks mathwonk ... given that you have outlined a very likely resolution of the issue, I will keep your ideas in mind and read onward ...

Thanks again for your help and support...

Peter
 
  • #4
mathwonk
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thanks Peter. Note that this definition allows one to say that any spliting field of a separable polynomial is a normal and separable extension, i.e. Galois, in most people's terminology. although be careful to check since I think some people just call these normal.
 
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Here's van der Waerden's definition which I have learnt:

##K \subseteq L## is a normal field extension (## \Leftrightarrow L ## is a Galois extension of ##K##), iff ##L## is algebraic over ##K## and each irreducible polynomial ##f(x) \in K[x]## that has one zero in ##L##, i.e. ##g(\alpha) = 0## for an element ##\alpha \in L##, completely splits into linear factors in ##L[x]##. For short:
##K \subseteq L## is normal ##=## ##K \subseteq L## is Galois ##=## algebraic ##+## one zero in ##L## implies all zeros in ##L##.

An element ##\alpha## which is a zero of an irreducible polynomial ##f(x) \in K[x]##, i.e ##f(\alpha)=0##, is called separable element with respect to ##K##, if ##f(x)## has only separated (simple) zeros. This means in a splitting field for ##f(x)## all factors are linear, that is ##(x-\beta)^2## or higher powers must not occur as factors. The irreducible polynomial ##f(x)## is also called separable polynomial, if all its zeros are separable. Elements that are not separable are called inseparable and likewise the polynomial.

Now an extension ##K \subseteq L## is called separable extension with respect to ##K##, if it is algebraic and all elements of ##L## are separable with respect to ##K##.
Otherwise the extension is called inseparable.

A field ##K## is called perfect, if all irreducible polynomials in ##K[x]## are separable.

All fields of characteristic ##0## are perfect.

As a historic side note, van der Waerden mentions, that Steinitz originally called separable elements, polynomials and field extensions as "of first kind" and the others, the inseparable "of second kind". He (van der Waerden) further says: "... I suggest to call it separable as this is more illustrating because all zeros are separated."
 
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  • #6
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thanks Peter. Note that this definition allows one to say that any spliting field of a separable polynomial is a normal and separable extension, i.e. Galois, in most people's terminology. although be careful to check since I think some people just call these normal.

Thanks again ... yes, will carefully check ...

Peter
 
  • #7
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Here's van der Waerden's definition which I have learnt:

##K \subseteq L## is a normal field extension (## \Leftrightarrow L ## is a Galois extension of ##K##), iff ##L## is algebraic over ##K## and each irreducible polynomial ##f(x) \in K[x]## that has one zero in ##L##, i.e. ##g(\alpha) = 0## for an element ##\alpha \in L##, completely splits into linear factors in ##L[x]##. For short:
##K \subseteq L## is normal ##=## ##K \subseteq L## is Galois ##=## algebraic ##+## one zero in ##L## implies all zeros in ##L##.

An element ##\alpha## which is a zero of an irreducible polynomial ##f(x) \in K[x]##, i.e ##f(\alpha)=0##, is called separable element with respect to ##K##, if ##f(x)## has only separated (simple) zeros. This means in a splitting field for ##f(x)## all factors are linear, that is ##(x-\beta)^2## or higher powers must not occur as factors. The irreducible polynomial ##f(x)## is also called separable polynomial, if all its zeros are separable. Elements that are not separable are called inseparable and likewise the polynomial.

Now an extension ##K \subseteq L## is called separable extension with respect to ##K##, if it is algebraic and all elements of ##L## are separable with respect to ##K##.
Otherwise the extension is called inseparable.

A field ##K## is called perfect, if all irreducible polynomials in ##K[x]## are separable.

All fields of characteristic ##0## are perfect.

As a historic side note, van der Waerden mentions, that Steinitz originally called separable elements, polynomials and field extensions as "of first kind" and the others, the inseparable "of second kind". He (van der Waerden) further says: "... I suggest to call it separable as this is more illustrating because all zeros are separated."



Thanks fresh_42 ... that clarifies the issue considerably...

Peter
 

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