All Questions

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A question regarding finite field extensions

If I understand correctly, the definition of the degree of a field extension $L/K$ is the dimension of $L$ over $K$ interpreted as a vector space. Now if the degree is $n < \infty$, the basis looks ...
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What is the term for the mathematical relationship between $\mathbb{Z}_n$ and $\mathbb{Z}$?

...and if it's important, do those ideas have any generalization to more "exotic" number systems? The motivation for my question comes from reading some of the excellent answers posted to other ...
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Using Central Limit Theorem

Can anyone help me with it: Using the central limit theorem for suitable Poisson random variables, prove that $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}=1/2$$ Thanks!
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Does Fermat's Last Theorem hold for cyclotomic integers in $\mathbb{Q(\zeta_{37})}$?

The first irregular prime is 37. Does FLT(37) $$x^{37} + y^{37} = z^{37}$$ have any solutions in the ring of integers of $\mathbb Q(\zeta_{37})$, where $\zeta_{37}$ is a primitive 37th root of ...
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Finding Characteristic Function

Can anyone help me with this problem? The random variable $X_n$ takes the values $\frac{k}{n}$, $k=1,2,\ldots,n$, each with probability $\frac{1}{n}$. Find its characteristic function and the limit as ...
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Examples of function sequences in C[0,1] that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
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Minimal polynomials

Can someone tell me if this is right: I would like to find the minimal polynomial of (i) $\sqrt[4]{2}i$ over $\mathbb{Q}$ (ii) $\sqrt[4]{2}i$ over $\mathbb{R}$ (i): $\sqrt[4]{2}i$ is a root of ...
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costructing a diffeomorphism

Let $i:N\to M$ be a smooth embedding, $\pi:E\to N$ a vector bundle and $s_0:N\to E$ is its zero section. I have an open neighborhood $U$ of $s_0(N)$ in $E$, and $f:U\to M$ is a smooth map such that ...