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Let $K$ be a field , let's consider the field of rationals functions over x , $k(x)$. Let $t\in k(x)$ be the rational function $\frac{p(x)}{q(x)}$ , where $P,Q$ have no common factors. I have to prove that the extension of fields $ K(t) \subset K(x) $ has degree the max degree between $P,Q$. I have to prove that the minimal polynomial of $x$ over the field $K(t)$ is the polynomial, $P(X)-tQ(X) \in (K(t))[x]$. Well Clearly $x$ is a root of the polynomial, so it remains to prove that it's irreducible.

First for Gauss lemma , the polynomial is irreducible in $(K(t))[x]$ iff is irreducible in $(K[t])[x] = (K[x])[t]$.

That consideration it's from a hint of the book ( Dummit Section 13.2 , Exercise 18). I suppose that if exist a factorization, then $P,Q$ will have common factors. But I don't know how to work here, it's very confusing. Please help me. $$ \eqalign{ & \left( {\sum\limits_{k = 0}^n {a_k (x)t^k } } \right)\left( {\sum\limits_{j = 0}^m {b_j (x)t^j } } \right) = P(x) - tQ(x) \cr & a_k (x),b_j (x) \in K[x] \cr} $$ I don't know if the possible factorization will be in $t$ or in $x$ I'm very confused right now. Sorry for being so stupid

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1 Answer 1

up vote 2 down vote accepted

Hint: It's a linear polynomial in $t$. When is a linear polynomial irreducible?

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I don't know what do you mean, I'm searching for divisors, what polynomial is linear? –  Daniel Oct 1 '12 at 4:34
Sorry for being so stupid Brandon :/ –  Daniel Oct 1 '12 at 4:52
Linear polynomials are always irreducible. Yeah you are right, but I'm confused, I did never used the fact that $(P,Q)=1$ So maybe it's wrong –  Daniel Oct 1 '12 at 13:39
Linear polynomials are always irreducible over fields. Over an arbitrary domain a linear polynomial is irreducible iff you can't factor out any non-unit. Since $K[x]$ is not a field, you need to check that this is not possible. You will need to use that $(P,Q) = 1$. –  Brandon Carter Oct 1 '12 at 14:08
In a domain, an element $r$ is called irreducible if whenever $r = ab$, then either $a$ is a unit or $b$ is a unit. The definition in Dummit and Foote is given on p. 284. –  Brandon Carter Oct 1 '12 at 14:21

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