Define If you add the zero polynomial, p(x) = 0 to Q(x) it becomes a field, call

Question

Define If you add the zero polynomial, p(x) = 0 to Q(x) it becomes a field, called the field of quotients of polynomials with rational coefficients. You may assume that it is a field, since you have been assuming that the field axioms hold for it for many years now. However, with the definitions: r(x) = p(x)/q(x) Q(x) is "positive" the highest order coefficient in p(x)q(x) is positive, i.e a_mb_n > 0, and r_2(x) > r_1(x) r_2(x) - r_1(x) is positive, Q(x) becomes an ordered field. Prove that. Prove that, if r(x) and w(x) are in Q(x) and r(x) > 0, there may be no n N such that nr(x) > w(x). In other words Q(x) is not archimedean.

Explanation / Answer

As r(x) and w(x) are in Q(x) and r(x) > 0.

Then for n in natural numbers,

we will have nr(x) and w(x)

As both r(x) and w(x) is in Q.

So, nr(x) - w(x) will be > 0

But its infinitesimal in the field.

So, no matter how much big is n,

it is not archimedean.

Hence Proved

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.