Intereting Posts

$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ is that correct?
$\Delta^d m^n =d! \sum_{k} \left { {k+n} \brace m + d}(-1)^{m+k}$ Is this a new formula?
How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
prove that the greatest number of regions that $n \geq 1$ circles can divide the plane is $n^2-n+2$
integral involving log poisson-like and rational
Definite integral of Normal Distribution
Calculus conjecture
Intuitive understanding of the Reidemeister-Schreier Theorem
How to derive the proximal operator of the Euclidian norm?
Understanding Reed-Solomon as it applies to Shamir secret sharing
Locally compact nonarchimedian fields
Help solving recursive relations
Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$.
There are compact operators that are not norm-limits of finite-rank operators
Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}$

Let $F$ be a field whose characteristic is $\neq 2$. Suppose the minimum polynomial of $a$ over $F$ has degree $2$. Prove that $F(a)$ is of the form $F(\sqrt{b})$ for some $b\in F$.

Well, $F(a)$ consists of all $x+ya$ for $x,y\in F$. I don’t understand how we can find such $b$.

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- Cogroup structures on the profinite completion of the integers
- Showing field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}$ degree 8
- Degree of splitting field less than n!
- Hint on an exercise of Mathieu groups
- Let $D$ be a UFD. If an element of $D$ is not a square in $D$ then is it true that it is not a square in the fraction field of $D$?

- The Center of $\operatorname{GL}(n,k)$
- units of a ring of integers
- For polynomial $f$, does $f$(rational) = rational$^2$ always imply that $f(x) = g(x)^2$?
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- Exercise from Serre's “Trees” - prove that a given group is trivial
- Examples of non-isomorphic fields with isomorphic group of units and additive group structure
- Prove that a nonzero integer p is prime if and only if the ideal (p) is maximal in Z.
- Why Do Structured Sets Often Get Referred to Only by the Set?
- A ring that is not a Euclidean domain
- Order of elements in $Z_n$

Hint: what is the formula for the solution of the second degree equation?

Try to first find $b$ such that $a \in F(\sqrt{b})$. Explain why $F(a)=F(\sqrt{b})$.

Where did the $char \neq 2$ become important?

**Hint** $\ 2\ne 0\,\Rightarrow\ 1/2\in F\,$ which enables one to complete the square and use the quadratic formula, yielding a root of the form $\, x = a + \sqrt{b}\,$ for $\,a,b\in F.$

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