{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

I am placing another link to something I just uploaded to cocalc (the incorporated  incarnation of SageMath).

I fully accept that this may be posted just for me and I do not demand nor request anyone else take an interest. 

It's just that I forced myself to discover a way to do something with Sage (computer algebra system) that had been very mysterious to me for the longest time.    All such breakthroughs are worthy of being posted here since, well, there will be times when I might appreciate having done so (in the future if I were to lose my physical notebooks or flashdrives).

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Notice that you discuss the reducibility or factorization of a polynomial only with reference to a specific field containing the coefficients of the polynomial.  You can easily find polynomials that are irreducible over one field, but reducible over another field.

I found this tactic in Chapter 7 (p.138) of Computational Mathematics with SageMath, a $70 unaffordable book which is nonetheless made available by the generous authors in pdf format.

Here is a link to the uploaded file at cocalc:  Factorization of Polynomials

Across the pond they use "real English," spelling factorization with an S, but in these states in north Amerika, we spell it with a Z.   If you spell the English way in school while in Amerika, it is marked as incorrect.   

No, I'm not familiar with modal logic; but I did take an intrest in fuzzy logic back in the 1990's.  It's also a non-classical logic with qualifiers between 0 and 1.

If the air is very hot, then run the air-conditoner fast.
If the air is cool, turn air-conditioner off.
If the air is warm, then run the air-conditoner at normal speed.
etc ....

The more if .. then statements, the more intelligence.

Mathematics is vast.  One of the challenges is staying focused in the area of interest.

Logic is the oldest link, as far as where mathematics stands with regards to philosophy.  I understand why this modal logic might appeal to you, since it appears to get into enriching the language to capture some of the more subtle "human" qualifiers and metrics.

I am sure that I have had to defer a vast amount of material to the shelves just to remain as focused as i have these past few years.   I am hoping to master some strong foundations so that, just in case I live to be much older, I might tinker around with physics with as strong a mathematical tool-kit as possible ... as an old man living on beans, almond butter, custom made flat breads and thin-crusted pizzas.

Just as Husserl employed the epoche to block out the entire world so as to focus on the "lifeworld" (representation of the world as phenomena), I must LIMIT THE CONTENTS OF CONSCIOUSNESS so as not to drown in a sea of chaos.

Eventually I wish to spend several years exhausting "Concrete Mathematics" (mathematical foundations of computer science), but I feel this devotion to the more abstract material to be a once in a lifetime opportunity; hence I am stubbornly keeping my attention focused on specific texts. 

One of the challenges of remaining committed to this discipline is knowing where to start.  It's quite personal, isn't it?   I feel we are each on our own in coming to terms with just where we are at in our quest for mathematical maturity or even simply mathematical development.   I would not be the one to discourage you from exploring what interests you. 

I have left a trail as far as describing what interests me.