Irreducibility of the polynomial $f (x) = x^{2p} − x ^p + t$

Question

If $k$ is a ﬁeld of characteristic $p > 0$ and $f (x) = x^{2p} − x ^p + t \in k(t)[x]$, how can we show that $f (x)$ is an irreducible polynomial in $k(t)[x]$ and that $f (x)$ is inseparable?

If $f(x)$ is irreducible then $D_xf(x) = 0 \implies( f, D_xf) = ( f, 0) = f$. Hence, if $f(x)$ is irreducible then $( f, D_xf) \neq 1 \implies f(x)$ is inseparable.

Homework problem? If it is, people will not go beyond giving you a hint. — Lubin, Mar 04 '13 at 15:34

score 2 · Answer 1 · answered Mar 05 '13 at 03:03

2

$K$ is a field $\implies K[t]$ is a U.F.D.

By gauss lemma: $f(x)$ is irreducible in $K(t)[x] \iff f(x)$ is irreducible in $K[t][x]$

then $K[t][x]= K[x][t]$

$f(x)$ is irreducible in $K[x][t]$ being a linear polynomial in $K[x][t]$

hence $f(x)$ is irreducible in $K(t)[x] $

answered Mar 05 '13 at 03:03

jim

3,624

Nice answer. In Gauss Lemma, $f$ should be primitive. – Mar 05 '13 at 20:14
Very nice indeed, and much quicker than what I had in mind! – Lubin Mar 08 '13 at 20:17

Lubin · Accepted Answer · 2013-03-04T16:53:59.967

1

Here’s a hint: notice that $k(t)=k(\tau)$ when $\tau=1/t$. If $\rho$ is a root of $f$, what polynomial in $k(\tau)[x]$ does $1/\rho$ satisfy?

edited Mar 04 '13 at 16:53

answered Mar 04 '13 at 15:50

Lubin

62,818

sir i have posted an answer could you tell me is it right? – jim Mar 05 '13 at 03:05

Irreducibility of the polynomial $f (x) = x^{2p} − x ^p + t$

2 Answers2