Linked Questions

1
vote
1answer
118 views

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. ...
38
votes
12answers
15k views

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?
27
votes
7answers
39k views

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 $a_0,a_1,\ldots,a_n$ ...
25
votes
5answers
19k views

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 ...
19
votes
5answers
3k views

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 ...
10
votes
2answers
2k views

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 $char(F)=2$?
6
votes
5answers
2k views

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 ...
3
votes
4answers
1k views

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 ...
4
votes
2answers
248 views

Number of roots of polynomials in $\mathbb Z/p \mathbb Z [x]$

I was given this proposition but I was never able to prove it. Does anyone know how to solve this? Let $f$ be a polynomial in $\mathbb{Z}/p\mathbb{Z}[x]$, where $p$ is prime. Then $f$ has at most $\...
1
vote
4answers
151 views

Why can $(\lambda - a)(\lambda - d)-bc = 0$ be rewritten as $(\lambda - \lambda_1)(\lambda - \lambda_2) = 0$?

I have $(\lambda - a)(\lambda - d)-bc = 0$ which can also be written as $\lambda^2 - \lambda(a+d) + (ad - bc) = 0$. This quadratic equation can be solved by finding the roots $\lambda_1$ and $\...
3
votes
1answer
215 views

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 $f=g \iff f(x)= g(x), \forall x \in R$. $\bf Attempt:$ ...
2
votes
5answers
92 views

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
1
vote
2answers
90 views

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 ...
1
vote
2answers
65 views

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 , $a^{2019} \equiv a^{675} \mod 2019$ , ...
0
votes
2answers
106 views

Classes that are their multiplicative inverses

My teacher has demonstrated this problem at class but I didn't understand at the time, and now I can't find any material in internet about this specific problem: "How can I prove that in the finite ...

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