As for nonzero polynomials not having zeroes everywhere, I guess the problem (for integral domains) is that it would have to have a factor of $(x-k)$ for all integers $k$
(This breaks if you break either the integral domain part or the characteristic zero part)
So, consequently, $\forall x,f(x)g(x)=0$ would then be the same as $fg=0$; $(\forall x,f(x)=0)\lor(\forall x,g(x)=0)$ would be the same as $f=0\lor g=0$. And the equivalence of $fg=0$ and $f=0\lor g=0$ comes from the integral domain-iness of the polynomial ring.
Right, that you can't have $f(x) = 0$ on $R$ but $f \neq 0$ is just from being an integral domain. Just factor out roots one after another (which you can do, 'cuz, integral domain)
Yeah, normally that's what I'd have thought, but I guess this book didn't care about non-separable Hilbert spaces so it just said everything is. ('-')\
(Disclaimer: I only know what that word means because the last chapter in my analysis book from first quarter was about Fourier analysis on such groups)
How does the proof of equivalence between the number theoretic definition and the other ones (any one you prefer - both of the other are geometric) go? Do you remember it off the top of your head?
@Daminark Lol I think this thing just proved P != NP. "The proof is trivial! Just view the problem as a sesquilinear ultrafilter whose elements are simplicial complexity classes." Too bad it lacks rigor (and correctness)
Which also explains where the term ideal comes from
You can see that these ideals are a substitution for numbers, they are ideal numbers in our domain, and prime ideals are the ideal primes in this domain.
Although ideal numbers was a different term invented byKummer
Wikipedia is interesting: The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory
I can prove Q(sqrt(-5)) is not principle (look at 2 x 3 = (1 + sqrt(5)) x (1 - sqrt(5)), so (2, 1 + sqrt(-5)) should work) but not that it's class no. 2
Quick question about quarternions and complex numbers. I can represent a quarternion as 4 numbers. Some computer languages have a built in complex number type. It might be faster to represent a quarternion as 2 complex numbers. Can I do that?
