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I understand construction of irreducible polynomials over finite fields is a non trivial problem. Can anyone please refer me to good resources regarding constructive approaches of irreducible polynomials?

Also a perspective of how construction of irreducible polynomial will help to understand other branches of mathematics such as coding theory, cryptography,etc., will be very helpful.

I am currently reading Lidl's 'Finite Fields' and found the conditions regarding the same problem a bit absurd looking and not very intuitive, (specially chapher 3, section 3 part). Can I have better ways to comprehend those results?

Added by mixedmath
The OP indicated in the comments that the parts of this book that are relevant are theorems 3.36 and 3.37. For ease, I display what is freely available here. Unfortunately, there is no theorem 3.36. But there is an example 3.36:

enter image description here

And the part of the theorem that we can access:

enter image description here
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enter image description here

The rest is not included in the preview.

Edit:

I found this http://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/1986a/art.pdf , in this paper the authors give an algorithm for constructing irreducible polynomials, but my mathematical background does not permit me to understand the paper fully, still if someone please explains the steps in the first algorithm with an example i will be glad.

Specifically the fourth step of the first algorithm remains unclear to me.

  • 1
    You cannot assume that everyone has a copy of your book at hand. What are the absurd unintuitive conditions in Lidl's book?2012-05-28
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    If you are refering to Lidl & Niederreiter, then I would say that it **is** the good resource. I know of relatively few families of irreducible polynomials, and most (if not all?) are listed in that book. Either buried in the theory, or as exercises. I understand you in the sense that that section may disappoint you at the first reading. You would hope someting better to be there, but no better recipes are known.2012-05-28
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    @jyrki latonen, thanks for the suggestion. But can you please elaborate on the thought process behind those results, specifically, if some one is not aware of the results before hand then how will he/she be inspired to think on those conditions and the techniques?2012-05-28
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    For the purposes of constructing fields of characteristic two (useful in many an application in your list) I use the primitive (hence irreducible) polynomials listed [here](http://web.eecs.utk.edu/~plank/plank/papers/CS-07-593/primitive-polynomial-table.txt).2012-05-28
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    @Phira unfortunately i can't copy the theorems and they are so long i can't reproduce them, will you be kind enough to check http://books.google.co.in/books?id=xqMqxQTFUkMC&pg=PR14&dq=Rudolf+Lidl++chapter+3&hl=en&sa=X&ei=5aXDT9LeJIvJrQff35y3CQ&ved=0CDUQ6AEwAA#v=onepage&q&f=false , theorem 3.36 and theorem 3.37? sorry for the inconvenience2012-05-28

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