At 11:00 - how do we know that p-q is irreducible? There's no guarantee that adding or subtracting irreducible equations results in another irreducible equation, right?
We have p-q=0=r Suppose r is not irreducible, that is, r=st Then 0=r=st st belongs to an integral domain, so either s=0 or t=0 This contradicts the minimality of the degree of r So, r is irreducible.
If p-q is reducible, then you can just factor something out and get a polynomial of even smaller degree, which would contradict the irreducibility of each of p and q to begin with.
Good description. If we state that we are in a principal ideal domain would this be simpler. There must be a minimal polynomial generator and must be unique because it is minimal and monic as here.
At 11:00 - how do we know that p-q is irreducible? There's no guarantee that adding or subtracting irreducible equations results in another irreducible equation, right?
I also wonder this. He just glossed over explaining that statement.
We have p-q=0=r
Suppose r is not irreducible, that is, r=st
Then 0=r=st
st belongs to an integral domain, so either s=0 or t=0
This contradicts the minimality of the degree of r
So, r is irreducible.
If p-q is reducible, then you can just factor something out and get a polynomial of even smaller degree, which would contradict the irreducibility of each of p and q to begin with.
Thank you for your clear expositions. I took college-level Abstract Algebra 40 years ago, and yet, these lectures are very enjoyable. Viva Galois!
Out of curiosity, did you use anything you learned in abstract algebra since taking that class many years ago?
Good description. If we state that we are in a principal ideal domain would this be simpler. There must be a minimal polynomial generator and must be unique because it is minimal and monic as here.
The fourth example, namely the sqrt(-1) over Z3, has min p = t^2 + 1 = t^2 - 2, seems to be not unique?
I think those are actually the same polynomials, since +1 = -2 in Z3.
Existence and uniqueness? More like "An amazing playlist this is!" 👍
100th like, great video!
One year later, I'm glad to have given this video its 115th like!
best explanation,,, thank you so much☺️
Did you really just use South Park
Yes, yes he did.