Difficult)))
That is insane that the rational root test and then the eisenstein criterion are true! the proof was very hard to follow, and I think I spent about 30 minutes working through each proof to show myself why it actually worked...but it is amazingly useful! Such detailed specifics are powerful tools for showing irreducible polynomials in Q[x].
Ahhh!!! conjugates of complex numbers...I had to run through a crash review on how operations work over complex numbers.
Reflective)))
These would have been loads of fun to use if I had known them in differential equations!!!
The proof of 4.22 uses many theorems we have in our toolbox. the definition of roots, factors, the rational root test, and the factorization of a reducible polynomial!! fun stuff!
It is amazing to me that by simply picking a p such that Zp[x] doesn't divide An then having the polynomial be irreducible in mod p also guarantees the original polynomial to be irreducible in Q[x] but I need to remember that reducible in mod p doesn't imply that it is reducible in Q[x].
BUT C[x] for any nonconstant polynomial is algebraically closed or is guaranteed to have a root and is irreducible if and only if the polynomial is degree 1.
AND FINALLY GET TO THE TOOLS I HAVE BEEN USING IN CALCULUS AND DIFF.EQU.!!!
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