page1913 Rings Matthias Lorentzen...mattegrisenforlag.com
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Question
Let `P(x) = a_n x^n + a_{n-1} x^{n-1} + ... `
`+ a_1 x + a_0` ,
`a_i in QQ` and `a_n ne 0`.
Now, suppose `P(pi^3) = 0` such that
`a_n (pi^3)^n + a_{n-1}(pi^3)^{n-1} + ...`
`+ a_1 pi^3 + a_0 = 0`.
We can rewrite each term `(pi^3)^k` as `pi^{3k}`, so
`P(pi^3) = a_n pi^{3n} + a_{n-1} pi^{3(n-1)} + ...`
`+ a_1 pi^3 + a_0 = 0`.
When considering this, is `pi^3` algebraic over `QQ`
?
