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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:
Bland defines separable polynomials as follows ... and also gives an example ...
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 2In 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 3In 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
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:
Bland defines separable polynomials as follows ... and also gives an example ...
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 2In 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 3In 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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Bland - Definition of a Separable Polynomial ....png13.4 KB · Views: 780
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Bland - Example of a Separable Polynomial ....png28 KB · Views: 697