Being away from web-logging for awhile, I found myself not intensively occupied in anything displace for, rather hesitant to write the thoughts and outcome down to public. Not rationally enough, I’ve been almost feeling like running out of words without “being run”, for reason that those matters in mind are what is like at the two poles; that would win the last importance for others.
Coming to the private work, the math ideas are just too trivial for many people; in contrary to the amount of time consumed, the rest out of which essential lives such as meals, bath, and work shall be entirely assigned to math; of course, in order to just know it.
What my telling is not about to complain (rather I love it thoroughly), only to say that starting to study math when he has achieved degree in totally irrelevant field brings about such a burden of isolation even when half a decade has passed.
The following math problem is what I thought of earlier.
Propositional: Let an arbitrary element f(T) of a polynomial field K[T] over K is irreducible, then f(T) is algebraically separable.
Proof: As to a polynomial F(T) irreducible over an integral domain R, let us use a contrapositive that if F(T) is not divided by g(T)∈R[T], then F(T) and g(T) are coprime. Adopting the method of reduction to the absurd, assume that f(T) is inseparable, now that:
gcd(f(T), f'(T))≠1
∴f(T) divides f'(T) (contrapositive)
∴f'(T)=0 (for which polynomial order comparison accounts)
∴f(T) is an invariable polynomial (for reason that char(K)=0)
∴opposed to the assumption that f(T) is irreducible polynomial.