## algebraic curves – Under what conditions is the compositum of two rational cubic Kummer extensions a rational function field?

Let $$k$$ be an algebraically closed field of characteristic $$p > 3$$ and $$F = k(x)$$ be the rational function field in the variable $$x$$. Consider two Kummer extensions $$F_1 = F(sqrt(3){g_1})$$, $$F_2 = F(sqrt(3){g_2})$$ of degree $$3$$, where $$g_1, g_2 in F^*$$. Suppose that $$F_1, F_2$$ are also rational, that is $$F_1 = k(x_1)$$, $$F_2 = k(x_2)$$ for some $$x_1 in F_1$$, $$x_2 in F_2$$.

Under what conditions is the compositum $$F_1F_2 = k(sqrt(3){g_1}, sqrt(3){g_2})$$ a rational function field?

## algebraic geometry – Plane minus a projective line vs. plane minus a conic

Let $$k$$ be a field. In $$mathbb{P} = mathbb{P}^2(k)$$, an irreducible conic $$C$$ is isomorphic to a projective line $$U = mathbb{P}^1(k)$$ (which we take to be a line in $$mathbb{P}$$). If I am not mistaken, we have that $$mathbb{P} setminus U$$ is birational to $$mathbb{P} setminus C$$. What is the easiest way to see this ?

## algebraic curves – A question about principal divisors and poles

Thanks for contributing an answer to MathOverflow!

But avoid

• Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. MathJax reference.

## Mathematica inverting algebraic output order

This is a somewhat silly requirement, but would help a lot in my workflow. Whenever I output an algebraic expression which contains a number, mathematica always orders the number first whenever possible. For example, if I input

x+1


Mathematica will output

1+x


This repeats itself in other situations which I would prefer not to, as for example when writing to subscripts. Example:

Subscript[x, n] + 1


Outputs to

1+ Subscript[x, n]


Is there a way to prevent this or guide mathematica as to the wanted output? It happens often also in algebraic manipulations that mathematica alters the order of commutative operations.

Thanks

## algebraic geometry – Are any two points of a variety contained in an open affine subset?

In Mumford’s Red Book of Varieties and Schemes, to prove that a projective variety is separated, they use a lemma in which they prove it is enough to show that every two points are contained in an open affine subset.

My question is: does it exist any (separated) variety and two points on it that are not both contained in an open affine subset? If such an example exists, can it be quasi-affine?

## algebraic geometry – Definition of locally generated by $mathscr{S}_1$ as an $mathscr{O}_X$-algebra

In Hartshorne’s “Alegebaic Geometory” 2.7,p160 (Dagger)

$$mathscr{S}$$is generated by $$mathscr{S}_1$$ as an $$mathscr{O}_X$$-algebra. “

but, I can’t understand mean of “locally”in this case.
Please tell me exact definition. Thanks.

## quadratic forms – Do we include operators before the coefficient when applying algebraic formulas to equations?

I do some simple retraining on college algebra after a long time, but I have come to find some inconsistencies in my understanding of how to apply the formulas, do we include operators before the coefficient when applying algebraic formulas to equations?
I fight in particular with these two formulas:

2x²+14x-196 = 0, when applied to the formula (-b±√(b²- 4ac))/2a, gives (-14±√((14)²-4(2)(-196)))/2(2). The answer here is x = -14, x = 7 checked.

Note that the coefficient "c" must take the negative operator to give the answer.

However:
If we are consistent and say yes, operators must be included in formulas:
When using the formula (what is this algebraic formula called btw?) (a-b)² = a²-2ab+b², when applied to (2-3)², gives: 2²-2(2)(-3)+(-3)² = 4+12+9=25. Which is clearly the wrong answer.

I'm struggling with this inconsistency, can anyone help me?

## algebraic geometry – T / F: finite module on reduced Noeterian ring whose reduction to each maximum ideal has the same dimension => locally free module?

In Hartshorne's algebraic geometry, there is an exercise (exercise 2.5.8 (3)): ($$Frac$$ means fraction field of an integral domain)

Assume $$X$$ is a reduced (noeterian) diet (WLOG $$X = Spec , R$$), and suppose $$M$$ is a coherent (i.e. qcoh + finite generated) $$mathcal O _X$$– (WLOG $$R$$-) module, with the property that,
$$varphi ( mathfrak p) = dim _ {Frac , ( mathcal O_ {X, p} / mathfrak p)} M _ { mathfrak p} otimes _ { mathcal O_ {X, p }} Frac , ( mathcal O_ {X, p} / mathfrak p)$$

is constant for all $$mathfrak p$$ in $$Spec , R$$. We would like to show $$M$$ is a local for free $$mathcal O_X$$-module.

This exercise is not too difficult when switching to the total fraction ring of affine $$R$$ and check what's going on at the generic point of each component, then use Nakayama; but which uses the theoretical generic points of the diagram harshly.

But this question seems to have a rather interesting meaning even for classical algebraic varieties (or arithmetic varieties, for example integer rings or Neron models) and in the more classical perspective, we only discuss the maximum spectra ; but in this case we cannot use the above argument (the condition that $$varphi ( mathfrak m)$$ is constant on the max spectrum is now lower than above!). I still believe that the result should always be true, so I would like to know proof of it.

## algebraic number theory – Counting prime ideals in cyclotomic integers

Consider the ring of cyclotomic integers $$mathbb {Z} (e ^ {2 pi i / p})$$ and consider a prime integer q such that q is not congruent to 1 mod p. How do I determine the number of prime ideals containing $$(q)$$? For the case $$p equiv 1 mathrm {mod}$$ $$p$$, I have seen people use the cyclotomic polynomial but I am confused on how to do it when q is not congruent to 1 mod p. Can someone provide an example to illustrate the procedure, such as $$mathbb {Z} (e ^ {2 pi i / 7})$$ with q $$= 11 equivalent 4$$ mod $$7$$?

## abstract algebra – If $alpha$ is algebraic on $overline { mathbb {Q}}$ then $alpha in overline { mathbb {Q}}$

I find it hard to show that if $$alpha$$ is algebraic $$overline { mathbb {Q}}$$ so $$alpha in overline { mathbb {Q}}$$. This is what I have done so far, it is likely that I have taken a completely wrong direction.

I know the whole $$F = { beta in mathbb C: beta text {is algebraic on} overline { mathbb {Q}} }$$ must be a field, so if $$alpha in F ^ { times}$$ so $$alpha ^ {- 1} in F$$. In particular $$alpha not in overline { mathbb {Q}}$$ if and only if $$alpha ^ {- 1} not in overline { mathbb {Q}}$$.

Let $$f in overline { mathbb {Q}} (X)$$ satisfied $$f ( alpha) = 0$$. so $$f ( alpha) = a_n alpha ^ n + points a_1 alpha + a_0 = 0$$ or $$a_i in overline { mathbb {Q}}$$. Let $$g$$ to be the minimal polynomial of $$-a_0$$ more than $$mathbb Q$$. So $$g (f (x) -a_0)$$ is a polynomial with $$alpha$$ as a root and with its constant coefficient $$mathbb Q$$. Together $$f_2 (x) = frac {g (f (x) -a_0)} {x}$$ and we have a rational function of the form $$b_ {m} x ^ m + b_ {m-1} x ^ {m-1} + points + b_0 + b _ {- 1} x ^ {- 1}$$ or $$m = n deg (g)$$ and $$b _ {- 1} in mathbb {Q}$$.

If we consider a polynomial $$g_2 (x) in mathbb Q (X)$$ such as $$g_2 (-b_0) = 0$$ we can keep repeating this process until we get a rational function $$h (x) = c_ix ^ i + c_ {i-1} x ^ {i-1} + dots + c_1x + c_0 + c _ {- 1} x ^ {- 1} + dots c _ {- j} x ^ {- j}$$ who satisfies $$h ( alpha) = 0$$ and or $$c _ {- 1}, dots, c _ {- j} in mathbb Q$$.

If this process finally ends (which would require that all the coefficients be in $$mathbb Q$$) then $$alpha ^ {- 1} in overline { mathbb {Q}}$$ which would imply $$alpha in overline { mathbb {Q}}$$.

However, I am not sure that this process will always end.