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### Self Inverse in Integral Domains [duplicate]

I need to show that if R is an integral domain and it has unity then the only elements of R which are inverse to itself are 1 and -1 (with respect to multiplication). But I don't know where to start. ...
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### Is There A Polynomial That Has Infinitely Many Roots?

Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?
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### How to prove that a polynomial of degree $n$ has at most $n$ roots?

How can I prove, that a polynomial function $$f(x) = \sum_{0\le k \le n}a_k x^k\qquad n\in\mathbb N,\ a_k\in\mathbb C$$ is zero for at most $n$ different values of $x$, unless all $a_0,a_1,\ldots,a_n$ ...
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### Simple explanation and examples of the Miller-Rabin Primality Test

Coming from an understanding of Fermat's primality test, I'm looking for a clear explanation of the Miller-Rabin primality test. Specifically: I understand that for some reason, having non-trivial ...
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### How does partial fraction decomposition avoid division by zero?

This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example: $$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$ Multiplying ...
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### Finite field, every element is a square implies char equal 2

If $F$ is a finite field such that every element is a square, why must $char(F)=2$?
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### Why is “division by $(z-1)$” valid here?

Is there an easy way to justify: $$x(x-1)(x+1) \equiv x(x^2-1) \Rightarrow (x-1)(x+1) \equiv x^2-1,$$ even for $x=0$? I seemingly have to divide by $x$ which should place the restriction $x \neq 0$ on ...
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### Polynomial equal to polynomial of lower degree

I am studying Linear Algebra Done Right, chapter 2 problem 6 states: Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite ...
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### In $R[x]$, $f=g \iff f(x)=g(x), \forall x \in R$

Let $R$ be an integral domain and $R[x]$ the polynomial ring over $R$. Let $f,g \in R[x]$ such that $\max(\deg f, \deg g)< \#R$. Show that $f=g \iff f(x)= g(x), \forall x \in R$. $\bf Attempt:$ ...
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### How can you say it is true or false?

Can we compare an equation like I did. Given that $p(x)$ has real roots. If it true, Would it also be true when roots are imaginary? Is it true. Please help
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### Is $f(a)\!=\!0\!=\!f(b)\Rightarrow (x\!-\!a)(x\!-\!b)\mid f(x)\,$ true if $\,a=b?\$ [Double Factor Theorem]

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
### Sum of the first four positive integer $\ a \$ such that $\ a^{2019} \pmod{2019 }= 1$
I've tried many times via Euler's Theorem. $\varphi( 2019) = 2\times 672 = 1344$ $a^{\varphi(2019)} \equiv a^{1344}\equiv 1 \mod 2019$ then , $a^{2019} \equiv a^{675} \mod 2019$ , ...