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Feb
11
awarded  Popular Question
Feb
8
accepted When does the cancellation law hold for the ring of polynomials over a field?
Feb
8
asked When does the cancellation law hold for the ring of polynomials over a field?
Feb
8
accepted Is there a notation for polynomial division?
Feb
8
asked Is there a notation for polynomial division?
Feb
7
accepted If $ R $ is a commutative ring with unity, then how do I prove: $ a \neq 0, ~ b \neq 0 \Longrightarrow a \cdot b \neq 0 $?
Feb
7
comment If $ R $ is a commutative ring with unity, then how do I prove: $ a \neq 0, ~ b \neq 0 \Longrightarrow a \cdot b \neq 0 $?
@Brian Do you mean, for any polynomial ring $R[X]$, $\deg f(X) + \deg g(X)$ may not be equal to $\deg f(X)g(X)$, where they are polynomials? What would be the weakest algebraic structure which makes that possible?
Feb
7
comment If $ R $ is a commutative ring with unity, then how do I prove: $ a \neq 0, ~ b \neq 0 \Longrightarrow a \cdot b \neq 0 $?
•means multiplication here. I dunno how to write small circle in Latex..
Feb
7
asked If $ R $ is a commutative ring with unity, then how do I prove: $ a \neq 0, ~ b \neq 0 \Longrightarrow a \cdot b \neq 0 $?
Feb
5
comment How do I define Polynomial ring $R[x]$ over a commutative ring $R$ with unity?
Thank you. Just for clarity, isn't your argument exactly the same as that i posted?
Feb
5
accepted How do I define Polynomial ring $R[x]$ over a commutative ring $R$ with unity?
Feb
5
comment How do I define Polynomial ring $R[x]$ over a commutative ring $R$ with unity?
Aha! I wonder why author of my textbook didn't write that one simple word 'sequence'.. Thank you
Feb
5
asked How do I define Polynomial ring $R[x]$ over a commutative ring $R$ with unity?
Feb
4
comment How do i prove that any two distinct irreducible monic polynomials in a field are relatively prime?
@rschwieb Yes, I made that assumption. Thank you!
Feb
4
comment How do i prove that any two distinct irreducible monic polynomials in a field are relatively prime?
@rschwieb For example, $Z_2=\{0,1\}$ is a field if $1+1=0, 1+0=1, 0•0=0, 1•0=0$. Here, polynomials $f(x)=x^2 + x + 1$ and $g(x)=1$ are not equal polynomials, but they are equal functions.
Feb
4
comment How do i prove that any two distinct irreducible monic polynomials in a field are relatively prime?
@rschwieb I was confused with the notion "$=$". Is there a special notation to distinguish "Equivalence in $F[X]$" from "equivalence in $F$"? I don't understand why one does not use some equivalence notation such as $\equiv$ for polynomials..
Feb
4
accepted How do i prove that any two distinct irreducible monic polynomials in a field are relatively prime?
Feb
4
comment How do i prove that any two distinct irreducible monic polynomials in a field are relatively prime?
@Math This is an exercise in my book, so I'm trying to prove it myself. There is no reference.
Feb
4
asked How do i prove that any two distinct irreducible monic polynomials in a field are relatively prime?
Feb
4
accepted Is there a ring which doesn't satisfy Division Algorithm for polynomials?