- #1
jostpuur
- 2,116
- 19
How do you prove that there does not exist numbers [itex]a,b\in\mathbb{Q}[/itex] such that
[tex]
0 = a + b\sqrt[3]{2} + \sqrt[3]{2}^2
[/tex]
[tex]
0 = a + b\sqrt[3]{2} + \sqrt[3]{2}^2
[/tex]
jostpuur said:I think we only know that [itex]X-\sqrt[3]{2}[/itex] must divide [itex]X^2+bX+a[/itex].
[itex]X^2+bX+a[/itex] doesn't need to divide anything.
micromass said:The polynomial [itex]X^2+aX+c[/itex] will have to divide [itex]X^3-2[/itex] in that case.
Norwegian said:What is the logic behind this statement? From a false premiss you can of course deduce anyhing you like, but I assume that is not your intended logic here. When it comes to minimal polynomials and such, the general result is usually the other way around: if g(x)[itex]\epsilon[/itex]Q[x] has a root [itex]\alpha[/itex], then the minimal polynomial of [itex]\alpha[/itex] divides g.
Norwegian said:What is the logic behind this statement? From a false premiss you can of course deduce anyhing you like, but I assume that is not your intended logic here. When it comes to minimal polynomials and such, the general result is usually the other way around: if g(x)[itex]\epsilon[/itex]Q[x] has a root [itex]\alpha[/itex], then the minimal polynomial of [itex]\alpha[/itex] divides g.
The Qube root of 2 is the number that, when multiplied by itself three times, gives a result of 2. In other words, it is the number that satisfies the equation x3 = 2.
The zero of a second order polynomial is the value of x that makes the polynomial equal to 0. It can be found by solving the quadratic equation ax2 + bx + c = 0, where a, b, and c are the coefficients of the polynomial.
The Qube root of 2 is important in mathematics because it is an irrational number, meaning it cannot be expressed as a fraction of two integers. This number is also used in various mathematical proofs and calculations.
The Qube root of 2 can be approximated using various numerical methods, such as the Newton-Raphson method or the bisection method. It can also be found using a calculator or by referencing a table of values.
The Qube root of 2 can be expressed as the zero of a second order polynomial with coefficients 1, 0, and -2. This means that the Qube root of 2 is a solution to the equation x2 - 2 = 0. In general, the Qube root of any number can be expressed as the zero of a second order polynomial with the appropriate coefficients.