Intereting Posts

Is there a nonnormal operator with spectrum strictly continuous?
Group of Order $p^2$ Isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p}$ $\times$ $\mathbb{Z}_{p}$
for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$
Notation on proving injectivity of a function $f:A^{B\;\cup\; C}\to A^B\times A^C$
Find all positive integers $n$ such that $n+2008$ divides $n^2 + 2008$ and $n+2009$ divides $n^2 + 2009$
Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$
Does there exist an unbounded function that is uniformly continuous?
Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.
Find all positive integers $a, b, c$ such that $a^2+1$ and $b^2+1$ are both primes and $(a^2+1)(b^2+1)=c^2+1$
Average number of trials until drawing $k$ red balls out of a box with $m$ blue and $n$ red balls
Dominated convergence theorem with $f_n(x)=\frac{1}{\sqrt{2\pi}}e^{-\sqrt{n}x}\left(1+\frac{x}{\sqrt{n}}\right)^n\chi_{}$
Does $\int_{0}^{\infty}\frac{dx}{1+(x\sin5x)^2}$ converge?
Prove that $F(x,y)=f(x-y)$ is Borel measurable
Do inequations exist with congruences?
Proof ¬q → ¬p from premise p → q using deductive system& Modus ponens

I can’t think of a way to prove this, can anyone help?

EDIT: I noticed this is a really simple question, and the confusion I made came from not seeing what was false intuition and what was the real Algebraic structure I was studying, I’m just starting to learn Algebra. Thanks for the answers, and sorry for those who thought my question didn’t follow the standards it should have, I really didn’t mean to cause this negative impact.

- Construct a finite field of order 27
- Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
- Schröder-Bernstein for abelian groups with direct summands
- Is Pythagoras the only relation to hold between $\cos$ and $\sin$?
- How to prove $\mathbb{Z}=\{a+b\sqrt{2}i\mid a,b\in\mathbb{Z}\}$ is a principal ideal domain?
- Existence of subgroup of order six in $A_4$

- Why should I care about fields of positive characteristic?
- Cardinality of the quotient ring $\mathbb{Z}/(x^2-3,2x+4)$
- Fixed field of automorphism $t\mapsto t+1$ of $k(t)$
- Finite Groups with exactly $n$ conjugacy classes $(n=2,3,…)$
- Every prime ideal is either zero or maximal in a PID.
- Notation for fields
- The total ring of fractions of a reduced Noetherian ring is a direct product of fields
- Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
- Proof verification: $\langle 2, x \rangle$ is a prime, not principal ideal
- Classifying Unital Commutative Rings of Order $p^2$

$a^2-1=0$ is equivalent to $a^2-1=(a-1)(a+1)=0$ since the domain is integral, $a=1$ or $a=-1$.

For added insight, let’s do a slightly more general case. We prove that a nonzero quadratic polynomial $f(x)$ over a domain $D$ has at most two roots $\,\color{#c00}{a\neq b},\,$ by using the Factor Theorem twice, and using that $D$ is a domain, so a product of nonzero elements remains nonzero.

$\begin{eqnarray}\rm\:f(b)= 0 &\ \Rightarrow\ &\rm f(x)\, =\, (x\!-\!b)\,g(x)\ \ for\ \ some\ \ g\in D[x]\\

\rm f(a) = (\color{#C00}{a\!-\!b})\,g(a) = 0 &\Rightarrow&\rm g(a)\, =\, 0\,\ \Rightarrow\,\ g(x) \,=\, (x\!-\!a)\,h(x)\ \ for\ \ some\ \ h\in F[x]\\

&\Rightarrow&\rm f(x)\, =\, (x\!-\!b)\,g(x) \,=\, (x\!-\!b)(x\!-\!a)\,h(x)\end{eqnarray}$

Comparing degree shows $h(x) = c$ is constant, and $f\neq 0\,\Rightarrow\,c\neq 0.\,$ If $f$ had a third root $r$ then $(r\!-\!b)(r\!-\!a)c = 0$, contra each factor is nonzero hence so is their product, since $D$ is a domain.

**Remark** $ $ The proof immediately generalizes by induction to yield that a nonzero polynomial of degree $n$ has at most $n$ roots over a domain. In fact this is a *characteristic property* of domains, since if $R$ is not a domain then there are $a,b\neq 0$ with $\,ab = 0\,$ so $\,f = ax\,$ has $2$ roots, $\,x = 0,b.$

- Another interesting integral related to the Omega constant
- On Reshetnikov's integral $\int_0^1\frac{dx}{\sqrtx\ \sqrt{1-x}\ \sqrt{1-x\,\alpha^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{3}\,|\alpha|}$
- In $R$, $f=g \iff f(x)=g(x), \forall x \in R$
- Finding $\pi$ factorial
- An example of a regular function over an open set
- study a sequence for increasing/decreasing
- Determine the Set of a Sum of Numbers
- What is the Euler Totient of Zero?
- Vandermonde determinant by induction
- Partial sums of exponential series
- General Topology and Basis definition
- General Solution to $x^2-2y^2=1$
- Consider the following Sturm-Liouville problem
- $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
- Intuition behind normal subgroups