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

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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

Sorry for my bad English.
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:

Quadratic formula, operators included here:
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.

Thanks in advance to all.

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.