13
votes
1answer
247 views
+50

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi ...
1
vote
3answers
384 views

Polynomials with Integer Coefficients and irrational roots

Is there a polynomial with integer coefficients which has √2 +√7  as a root?
7
votes
4answers
469 views

Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
0
votes
0answers
467 views

The Conjugate Roots Theorem for Irrational Roots

The Conjugate Roots Theorem for Irrational Roots states that for a polynomial $f(x)$ with integer coefficients, if a root of the equation $f(x) = 0$ is expressed as $a+\alpha$, where $a\in\mathbb{Q}$ ...
1
vote
0answers
210 views

Third degree polynomial with integer coefficient and three irrational roots

There are some polynomial with the above characteristic, and real roots of such polynomials cannot be found using rational number theorem and irrational conjugate theorem. The example of such function ...
3
votes
3answers
78 views

Number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$?

The number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$ is (A)0 (B)2 (C)3 (d)4 Actually im a 10 class student i don't know any of it,but my elder brother(IIT Coaching) cannot ...
1
vote
2answers
355 views

Proof $ \sqrt{1 + \sqrt[3]{2}} $ is irrational using the theorem about rational roots of a polynomial

I'm having trouble with this specific problem at the moment. The theorem states that if $n/m$ is a rational root of a polynomial with integer coefficients, the leading coefficient is divisible by m ...
1
vote
1answer
314 views

how to prove cubic root of 25 is irrational

how to prove cubic root of 25 is irrational using mathematical induction? I've been trying to do it for hours but can't get it, help plz guys :S
4
votes
1answer
172 views

Is it assumable that $2^{1/12}$ is irrational because $2^{1/2}$ is?

I need to prove that $2^{1/12}$ is irrational but I need to connect this to $2^{1/2}$ being irrational. I know how to prove that $2^{1/2}$ is irrational, but can I assume that $2^{1/12}$ is irrational ...