- #1

fishturtle1

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- Homework Statement
- Let ##F## be a field with characteristic ##p## and let ##f(x) = x^p - a \in F[x]##. Show ##f## is irreducible over ##F## or ##f## splits in ##F##.

- Relevant Equations
- ##F## has characteristic ##p## means that ##p## is the smallest positive integer such that for all ##x \in F## we have ##px = 0##.

##f(x)## splits in ##F## means ##f(x) = c_0(x-c_1)(x-c_2)\cdot\dots\cdot(x-c_n)## where ##c_i \in F##.

##f(x)## is irreducible over ##F## means if ##f(x) = g(x)h(x)## in ##F##, then ##g(x)## or ##h(x)## is a unit.

Suppose ##f## is reducible over ##F##. Then there exists ##g, h \in F## such that ##g, h## are not units and ##f = gh##. If there exists ##b \in F## such that ##b^p = a##, then ##(x - b)^p = x^p - b^p = x^p - a##, using the fact that ##F## has characteristic ##p##. So, if such a ##b \in F## exists, then ##f## splits in ##F##. But I don't think we can guarantee ##b## does exist. And I realize I didn't really use the assumption that ##f## is reducible. How to proceed?

Does ##f## being reducible over ##F## somehow imply ##a## has a pth root in ##F##?

Does ##f## being reducible over ##F## somehow imply ##a## has a pth root in ##F##?