Intereting Posts

Isometry from Banach Space to a Normed linear space maps
The equivalence relation $(z_1, n_1)\sim(z_2, n_2) :\Leftrightarrow z_1 \cdot n_2 = z_2 \cdot n_1$
What good are free groups?
TicTacToe State Space Choose Calculation
Homology of a simple chain complex
irreducibility of $x^{5}-2$ over $\mathbb{F}_{11}$.
Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?
Evaluating this integral $ \small\int \frac {x^2 dx} {(x\sin x+\cos x)^2} $
Is there a vector space that cannot be an inner product space?
Solution to $e^{e^x}=x$ and other applications of iterated functions?
Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
If a function is integrable, then it is bounded
Elementary proof for $\lim_{n \to\infty}\dfrac{n!e^n}{n^n} = +\infty$
Let $f : \to \mathbb{R}$ be a continuous function and $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $f(c) = 0$.
What is the Arg of $\sqrt{(t-1)(t-2)}$ at the point $t=0$?

Suppose $R$ is a field and $f$ is a polynomial of degree $d$ in $R[x]$. How do you show that each coset in $R[x]/\langle f\rangle$ may be represented by a unique polynomial of degree less than $d$? Secondly, if $R$ is finite with $n$ elements, how do you show that $R[x]/I$ has exactly $n^d$ cosets?

- What's the smallest exponent to give the identity in $S_n$?
- Minimal polynomial of $\omega:=\zeta_7+\overline{\zeta_7}$
- Classification of indecomposable modules over a given ring
- The Maximum possible order for an element $S_n$
- Solving an equation with irrational exponents
- Left Invertible Elements of a monoid
- An ideal which is not finitely generated
- Find a polynomial $p(x,y)$ with image all positive real numbers
- Characteristic of a field is $0$ or prime
- How to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers?

We will use the following fact (it should have been proved in the above-mentioned class): Let $K$ be any field. Then $K[x]$ is an Euclidean ring, i.e., for every two polynomials $f,g \in K[x]$ such that $g \neq 0$, there exist (unique!) polynomials $q, r$ such that $f = g \cdot q + r$ and $\deg r < \deg g$, with $\deg 0 = -\infty$.

Now, let $\overline{g}$ be any coset of $R[x]/\langle f \rangle$, and $g$ its representative. By division with remainder, we have $g = f \cdot q + r$ with $\deg r < \deg f$ and … (fill in the gaps here).

For the second part, count the polynomials with degree $< d$.

**Hint** $\rm\ R[x]/(f)\:$ has a complete system of reps being the least degree elements in the cosets, i.e. the remainders mod $\rm\:f,\:$ which exist (and are unique) by the Polynomial Division Algorithm..

Therefore the cardinality of the quotient ring equals the number of such reps, i.e. the number of polynomials $\rm\in R[x]\:$ with degree smaller than that of $\rm\:f.$

**Remark** $\ $ This is a generalization of the analogous argument for $\rm\:\Bbb Z/m.\:$ The argument generalizes to any ring with a Division (with Remainder) Algorithm, i.e. any Euclidean domain.

- Show that $\frac{1}{1+x}H(\frac{x}{1+x})=\sum^\infty_{k=0}x^k$
- Limit $\lim_{x\to\infty}x\tan^{-1}(f(x)/(x+g(x)))$
- Find $n$ and $k$ if $\:\binom{n\:}{k-1}=2002\:\:\:\binom{n\:}{k}=3003\:\:$
- How to show that $10^n – 1$ is divisible by $9$
- Multiplying Binomial Terms
- How can we show that $\mathbb Q$ is not free?
- Over $\mathbb{R}$, if $Z(p') \subset Z(p)$ when does $p' \vert p$?
- Existence of submersions from spheres into spheres
- Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$
- Beautiful proof for $e^{i \pi} = -1$
- A polynomial determined by two values
- Can area be irrational?
- How to evaluate the limit $\lim_{x\to 0} \frac{1-\cos(4x)}{\sin^2(7x)}$
- How to prove that the uniform topology is different from both the product and the box topology?
- Manipulating Algebraic Expression