## How would I use Reduce to find the intervals between the roots of a rationals polynomial?

I’m given two poly1. I expanded both of them and got:

``````poly1 = Expand((x - 1.1) (x - 2.2)^2 (x - 3.3)^3)
``````
``````191.329 - 521.805 x + 566.607 x^2 - 316.778 x^3 + 96.8 x^4 - 15.4 x^5 + x^6
``````
``````poly2 = Expand((x + 1.3) (x - 2.5)^2  (x - 3.7)^3)
``````
``````-411.556 + 346.357 x + 86.9611 x^2 - 191.712 x^3 + 81.89 x^4 - 14.8 x^5 + x^6
``````

Then I evaluated:

``````rat = poly1/poly2
``````
``````(191.329 - 521.805 x + 566.607 x^2 - 316.778 x^3 + 96.8 x^4 - 15.4 x^5 + x^6) /
(-411.556 + 346.357 x + 86.9611 x^2 - 191.712 x^3 + 81.89 x^4 - 14.8 x^5 + x^6)
``````

However, I have to use `Reduce` to find out the intervals on which the rational polynomial `rat` is positive and the intervals on which `rat` is negative. I have to reduce both polynomials, but Mathematica said it’s insufficient.

This is one of the codes I tried:

``````Reduce(191.32858800000005` - 521.8052400000001` x + 566.6067` x^2 - 316.778` x^3 + 96.8` x^4 - 15.399999999999999` x^5 + x^6 > 0, x)
``````

I was just wondering what I’m doing wrong.

## nt.number theory – Can you explain in pictorial way why there’re much more irrationals than rationals despite they are both infinite in the common usual sense?

This question isn’t likely to stay open as a research-level question, but I will try answer anyhow:

Part of your confusion is the following statement: “despite the known fact that every irrational has 2 immediate neighboring rationals on both sides”. This is not true.

What is true is that between any two irrational numbers there is a rational number; in fact, there are infinitely many rational numbers between any two irrational numbers.

Similarly, there are infinitely many irrational numbers between any two rational numbers.

If your statement above were true, then you would be correct. However, as stated, it is false.

## nt.number theory – rank 1 evaluations which are not discrete on finite transcendental extensions of rationals

assume $$K = mathbb {Q} (X_1, dots, X_n)$$ is a purely transcendental extension of rationals on an infinitely indeterminate number. Can anyone give an example of rank $$1$$ evaluation on $$K$$ who fails to be discreet?

If not, is there a theorem which shows that such a rank $$1$$ should the assessment be discreet?

## Circular definition of the rationals. – Mathematics Stack Exchange

If we define rational numbers as

A rational number is a number that can be a fraction $$frac pq$$ of two integers p and q, with the denominator $$q$$ not equal to zero.

But the integers themselves are rational numbers of the form $$frac p1$$ or $$p$$ is an integer. So the definition becomes circular. How can we avoid this?

## Find Galois Group on Rationals

I need help to determine the Galois group of the polynomial division field $$x ^ 4-2x ^ 3 + 2x ^ 2 + 2x + 1$$. Can I use Sage or Gap for this purpose and how? I am not able to understand this one.

## number theory – Galois elements determined by an action on the roots of the rationals?

Can there be an element $$sigma$$ of $$Gal ( overline { mathbb {Q}} / mathbb {Q})$$, other than identity and complex conjugation, which is completely determined up to conjugation by its action on $$sqrt[n]r$$ for everyone $$r in mathbb {Q}$$ and all $$n in mathbb {N}$$?

I'm expecting no, but if I'm wrong, can the subgroup of such elements be given another characterization?