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Jan
30
comment Cosets in quotient rings over ideals of the form $(n,f(x))$
And even if you do have leading coefficient 1, it's still complicated. Suppose $p(x)=x^2+1$. Dividing by $p(x)$ gives a remainder $a+bx$. Now how do you divide this by $n$? The most straightforward thing to do is divide $a$ and $b$ separately by $n$ and replace them with their remainders. So $a$ and $b$ can be any integer from $0,1,\ldots,n-1$. This gives you the cardinality of the quotient ring ($n^2$), but it does not tell you the ring structure, which is a number theory question - it depends in a more complicated way on the prime factorization of $n$.
Jan
30
comment Cosets in quotient rings over ideals of the form $(n,f(x))$
If $p(X)$ does not having leading coefficient $\pm 1$, then it is not so clear what it means to divide a polynomial by $p(X)$ and take the remainder. (e.g. How do you divide $x^3+1$ by $2x^2$ in $\mathbb{Z}[x]$?)
Jan
30
answered Mutually exclusive AND independent event (help with examples)
Jan
30
answered Cosets in quotient rings over ideals of the form $(n,f(x))$
Jan
6
comment Implicit assumptions about axes
It doesn't matter. You're just rotating the entire diagram by 90 degrees but that won't change the volume.
Jan
5
comment How to identify an error in a proof?
@Timbuc The first line is the statement that we are trying to prove (which is of course false), and the rest is the (erroneous) proof.
Jan
3
comment Finding a value from 5 systems of equations of 5 variables(CHMMC 2014)
Nice solution! (The leading coefficient of $x^4$ in $P(x)$ is actually $\sum a_i$, but it turns out that's not relevant to your solution.)
Dec
31
comment Group of isometries is closed in $GL_{n+1}$
@learner If your argument was intended to show that Isom is closed in $GL_{n+1}$, then your argument is correct. I thought you were addressing the second part of the question, whether Isom is compact.
Dec
26
answered Group of isometries is closed in $GL_{n+1}$
Nov
16
answered Probability of four-of-a-kind flawed logic
Nov
6
answered Isomorphism of quotient fields
Nov
2
comment How does $\in$ behave with simple algebra dealing with sub gradients?
What do you mean if a subset has a "related inverse"? Usually the sum of 2 subsets is defined as $A + B = \{ a+b : a \in A, b \in B\}$. With this definition, there is never an inverse of a subset that has more than one element. If for example $A = \{1,2\}$, then $B = \{-1, -2\}$ is not an inverse of $A$. The set $A+B$ contains the element 1 + (-2) = -1, so $A+B$ is not the identity element with respect to addition of subsets (which would be $\{0\}$.)
Nov
2
answered How does $\in$ behave with simple algebra dealing with sub gradients?
Oct
24
comment Intuitive Meaning of Quotient Ring
This "treating elements of $I$ as though it were zero" also explains why an ideal is defined the way it is: because 0+0=0 (hence $I$ should be closed under addition) and $a \cdot 0 = 0$ (hence $a \in R, x \in I$ should imply $ax \in I$).
Oct
23
answered How to find $α^2(β^4 +γ^4 +δ^4)+β^2(γ^4 +δ^4 +α^4)+γ^2(δ^4 +α^4 +β^4)+δ^2(α^4 +β^4 +γ^4)$
Oct
21
answered If $G$ is simple, then $\epsilon \leq {v \choose 2}$ - Bondy/Murty - Graph Theory with Applications Page 4
Oct
20
revised What to do with a random variable when we know its mean and variance but does not know which distribution it is?
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