Linked Questions

1
vote
1answer
178 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. ...
44
votes
12answers
19k 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?
30
votes
7answers
54k 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 $a_0,a_1,\ldots,a_n$ ...
28
votes
5answers
20k views

Split $n$ into nontrivial factors via a nontrivial square-root of $1\!\pmod{\!n}$

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 ...
20
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 ...
11
votes
2answers
3k 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
3answers
5k views

modulo version of the quadratic formula and Euler's criterion

Use the modulo version of the quadratic formula and Euler's criterion to decide if the following has a solution or not. $2x^2+5x+8 \equiv 0\pmod{37}$ I'm not sure how I would use what was being ...
4
votes
2answers
287 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 $\...
2
votes
2answers
550 views

$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$

I got a question to show that : If $p$ is prime number, then $$x^p - x \equiv x(x-1)(x-2)(x-3)\cdots (x -(p-1))\,\,\text{(mod }\,p\text{)}$$ Now I got 2 steps to show that the two polynomials ...
2
votes
4answers
97 views

If $a$ $\in$ $\mathbb R$ and $a \neq 1$ satisfies $a.a=a$, prove that $a=0$. [closed]

If $a$ $\in$ $\mathbb R$ and $a \neq 1$ satisfies $a.a=a$, prove that $a=0$. Though I find this expression intuitively right, I find it hard to prove it by only using the field axioms of $\mathbb R$.
4
votes
3answers
116 views

Little Fermat equivalence $\!\bmod p\!:\, a^p\equiv a\!\iff \!a^{p-1}\equiv 1\,$ for $\,a\not\equiv 0$

In various texts, I have seen Fermat's Little Theorem presented as: $\forall a\in\mathbb Z, a\not\equiv 0\pmod{p}$ and prime $p$, $a^{p-1}\equiv1\pmod{p}$. On the other hand, in a reputable text, I ...
3
votes
1answer
309 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 $f=g \iff f(x)= g(x), \forall x \in R$. $\bf Attempt:$ ...
1
vote
4answers
177 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 $\...

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