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15h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
@GeorgesElencwajg Thank you.
15h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
@GeorgesElencwajg I put something trivial here :)
15h
answered Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
15h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
@GeorgesElencwajg No. ${}{}{}{}{}$
15h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
@GeorgesElencwaig I will try the case $p=5$
15h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
For $p\in \{2,3\}$, I am able to see this. I will try now the case $p=5$
15h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
I am unable to see immediately why $f$ is reducible implies that $f$ has a root in $F$. But, I will try to prove it.
17h
revised Infinite sets don't exist!?
edited tags
17h
comment Infinite sets don't exist!?
@wim But I find $3$ more intuitive than $\infty$
19h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
I suppose you are assuming that f is reducilbe (because it its irreducible we are done), now how do we deduce that $f$ has a root
19h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
I know this, but why does $f$ reducible imply that it has a root
19h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
Hi I dont get the hint.
19h
comment Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$
Is $F$ a finite field ?
19h
comment How to solve $\ln x+x=1$
@user144251 See the book " The Art and Craft of Problem solving"
20h
comment How to solve $\ln x+x=1$
@user144251 No, and I actually consider myself not good at all in hard math contests such as IMO
20h
answered How to solve $\ln x+x=1$
2d
comment Online pronunciation of mathematicians names
@WillJagy Ahhh OK :)
2d
comment Online pronunciation of mathematicians names
@WillJagy What does this mean ?
2d
comment Online pronunciation of mathematicians names
@WillJagy My name can just be pronounced just by pronouncing every letter appearing in my name in the order they appear in my name. Some names do not satisfy the property that my name satisfies
2d
comment Online pronunciation of mathematicians names
Some Wikipedia articles don't have an online pronounciation such as this one: en.wikipedia.org/wiki/Saharon_Shelah