# residocity of elements in an extension of $\mathbb{F}_p$

Consider the following code :-

p=10010113
F=GF(p)
R=PolynomialRing(F,'x')
f=x^5 + 3212480*x^4 + 5943978*x^3 + 1041193*x^2 + 3212605*x + 4505026
F1.<a>=F.extension(f)
R1=PolynomialRing(F1,'x')
f1=derivative(f(x),x)
b=R1(f1(a))
b=F1(b)


Now since b $in$ F1 . Therefore F1(b**p^5-1) should output 1, but I am getting this output

sage: F1(b**(((p^5)-1)))
/usr/lib/sagemath/local/bin/sage-ipython:1: RuntimeWarning: invalid value encountered in power
#!/usr/bin/env python
9745575*a^4 + 8100949*a^3 + 6855548*a^2 + 351457*a + 548263


Which is absurd !

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First, i guess that you write

sage: R.inject_variables()
Defining x


to let the Python name x point to the indeterminate x after defining R. Then, indeed, since $(F1\setminus \{0\},\times)$ is a group of order p^5-1, you should have $b^{p^5-1}=1$, but it is not what you are writing, some parentheses are missing, you should write:

sage: b^(p^5-1)
1


which works !

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