Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [fraction-field]

Apt for questions related to fraction field or field of fractions (the ring of fractions of an integral domain).

Fraction field or field of fractions is the ring of fractions of an integral domain. The field of fractions of the ring of integers $\Bbb Z$ is the rational field $\Bbb Q$.

The field of fractions of an integral domain $R$ is the smallest field containing $R$, since it is obtained from $R$ by adding the least needed to make $R$ a field, namely the possibility of dividing by any nonzero element.

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Does fraction have the same meaning as rational in number theory?

I'm unable to get the difference between fraction and rational, we say $\frac{a}{b}$ is rational number if a and b are two integer with $b\neq 0$, and we can say also $\frac{a}{b}$ is a fraction but i don't know any reason for that, my question here…
zeraoulia rafik
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How to prove a number system is a fraction field of another?

For example, how can I show that $\mathbb{Q}$ is the fraction field of $\mathbb{Z}$? Or that $\mathbb{C}$ is the fraction field of $\mathbb{R}$? I understand that $\mathbb{Z}$ is a subring of $\mathbb{Q}$ & each r in $\mathbb{Q}$ can be written as a…
BigD4J
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Deducing that $\mathbb Z[2 \sqrt{2}]$ is not a $UFD.$

Here is the question I am trying to understand the solution of the last part in it: Let $R$ be an integral domain with quotient field $F$ and let $p(x)$ be a monic polynomial in $R[x].$ Assume that $p(x) = a(x) b(x)$ where $a(x)$ and $b(x)$ are…
Intuition
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Does imply f and g are relatively prime in K[x], that f- yg is irreducible in K(y)[x]?

Problem: Suppose that $K$ is a field and that $f$ and $g$ are relatively prime in $K[x]$. Show that $f- yg$ is irreducible in $K(y)[x]$. My attempt: Consider the polynomial $f-gY\in (K[x])[Y]$. This polynomial is irreducible. Indeed, if it were not,…
3435
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Fraction field of (A[b])

Let $A \subseteq B$ be commutative rings and $b \in B$. Is it true that $Frac(A[b])=Frac(A)[b] ?$ I felt it was true that $Frac(\Bbb Z[i])=\Bbb Q[i]$ so I thought it was maybe true for other rings too but not completely sure about it.
Kilkik
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