p adic number theory – conjugation of maximal algebraic tori

Accept $ G $ is a connected, reductive algebraic group over a non-Archimedean local field $ F $which is divided over a finite extent $ E / F $,

I often see a result that says "everything is maximum $ F $-Tori are conjugated over $ E $", by which I understand the following: Let $ G (E) $ denote the $ E $-Dots of the algebraic group $ G $;; then for each maximum $ F $-tori $ T, T $ $ of $ G $is there $ x in G (E) $ so that $ T (E) = xT & # 39; (E) x ^ {- 1} $,

In addition, the definitions show that if $ T, T $ $ are maximum $ F $-tori from $ G $then there is an isomorphism of $ T (F) $ on to $ T & # 39; (F) $ which is defined via $ E $,

My question is: Can the isomorphism be assumed to be conjugation in the second statement (as in the first statement)? That means: it follows from these results that if $ T, T $ $ are maximum $ F $-tori in $ G $then it exists $ x in G (E) $ so that $ T (F) = xT & # 39; (F) x ^ {- 1} $?

Any help (including proof of the first statement) is greatly appreciated!

Ring Theory – Let $ I: = (3,1+ sqrt {5} i) $ and $ R: = mathbb {Z}[sqrt{5}i]$. Show that $ I $ is not maximal.

To let $ I: = (3,1+ sqrt {5} i) $ and $ R: = mathbb {Z} ( sqrt {5} i) $, Show that $ I $ is not maximal. My goal is to show that $ R / I $ is a field. So far I have the following:

begin {align *}
mathbb {Z} ( sqrt {5} i) / (3,1+ sqrt {5} i) & cong ( mathbb {Z} (X) / (x ^ 2 + 5)) / (3 , 1 + X, X ^ 2 + 5) / (X ^ 2 + 5)) \
& = mathbb {Z} (X) / (3,1 + X, X ^ 2 + 5) \
& = mathbb {Z} / 3 mathbb {Z} (X) / (1 + X, X ^ 2 + 2).
end {align *}

I can not see how you reduce $ mathbb {Z} / 3 mathbb {Z} (X) / (1 + X, X ^ 2 + 5) $ further. Ideally, if I could write $ X ^ 2 + 2 $ as a multiple of $ 1 + X $I would be ready.

abstract algebra – Be $ R = mathbb Z[x]$ is the ring of polynomials over $ mathbb Z $. Prove that the ideal $ (x, p) $ is maximal if and only if $ p $ is a prime.

To let $ R = mathbb Z (x) $ let the ring of polynomials be over $ mathbb Z $, Prove that is the ideal $ (x, p) $ generated by $ x $ and $ p $, Where $ 0 <p in mathbb Z $is a maximum ideal of $ R $ then and only if $ p $ is a prime number. For a prime number $ p $, identify the field $ R / (x, p) $,

Here is my attempt:

(1) Suppose that $ (x, p) $ is maximum.

Accept $ p $ is not great. Then $ p = nk $ for some $ n, k in mathbb Z $,

The elements of $ (x, p) $ are of the form $ p + p_ {1} x + p_ {2} x ^ 2 + … $ Where $ p_ {i} $ are a multiple of $ p $ for all $ i in mathbb N $,

$ p = kn $ So the elements also have the form $ kn + kn_ {1} x + kn_ {2} x ^ 2 + … $

But then $ (x, p) = (x, kn) subseteq (x, k) $ and $ (x, n) $,

But $ (x, p) $ is maximum. Contradiction.

In order to $ p $ is prime.

(2) Suppose now $ p $ is prime, and that $ (x, p) $ is not maximal.

So there is a maximum amount $ M $ so that $ (x, p) subsetneq M subsetneq R $,

So there is a polynomial $ P in M ​​$ Where $ P = n + Q $. $ p $ does not split $ n $, and $ Q $ is a polynomial.

In order to $ (x, p) + P $ is an ideal larger than $ (x, p) $,

But $ p $ is great and does not split $ n $, in order to $ gcd (p, n) = 1 $,

In order to $ 1 in $(x, p) + P$, and is therefore $R $. A contradiction.

In order to $ (X, p) $ must be maximum.

Therefore, $ (x, p) $ is a maximum ideal if and only if $ p $ is prime.

(3) The field $ R / (x, p) $ has elements of form $ (x, p) + P $, Where $ P in R, but P notin (x, p) $, All such $ P $ are of the form $ n + Q $. $ p $ does not split $ n $, and $ Q $ is a polynomial.

I … think that's it? I'm not sure about my evidence for (1) and (2), and I'm not sure what else I need for (3). Any advice would be appreciated!

complex analysis – evidence of the existence of a maximal analytical continuation of a holomorphic germ

The following happens Lectures on Riemann's surfaces by O. Forster:

I do not understand the second paragraph of the proof. Why is the card? $ F: Z to Y $ well defined if we choose another curve $ a $ to $ x $, we could get another one $ eta $

For the relevant definitions:

Proofing Techniques – Prove that every complete prefix-free language is maximal

I am practicing a problem where I have to prove that every prefix-free language is maximal.

I know that a prefix-free language A is a maximum if it is not a proper subset of a prefix-free language, where a prefix-free language is full if

$$ sum_ {x in A} 2 ^ {- | x |} = 1 $$

Also, I know that a language A 0 {0, 1} * is a prefix-free if no element of A is a prefix of another element of A and that the force inequality says that for each prefix-free language A .

$$ sum_ {x in A} 2 ^ {- | x |} leqslant 1 $$

I'm pretty sure that a full free prefix language is maximal because it belongs to the maximum prefix-free set. But I do not know how to prove it formally. Should I deal with Kraft's inequality and what I know about the relationship between maximum and prefix-free quantities?

Any help would be appreciated!

ag.algebraic geometry – Are maximal tori conjugated locally localizable?

To let $ S $ to be and leave a scheme $ G to S $ to be a reductive group scheme. Then $ G $ admits that there is a maximal torus etale-local, and two maximal tori are conjugate etale-local, according to Theorem 3.2.6 and Corollary 3.2.7 in [B. Conrad, Reductuve group schemes] http://math.stanford.edu/~conrad/papers/luminysga3.pdf.

As stated in the book under Proposition 3.1.9 and at the beginning of Chapter 4, the presence of maximum Tori Zariski locally can be achieved (cf. [SGA3, XIV, 3.20]). My question is the following

To let $ T $ and $ T # $ be maximum tori $ G $ over defined $ S $are they conjugated Zariski-local?

In my case, $ T $ is split $ S $, and $ S $ is affine with $ Pic (S) = 0 $,

I would appreciate comments!

Determine the probability that X + Y is maximal for a given X

Consider some random variables $ X_1 $ , , , $ X_n $ they are independent and distributed identically to CDF $ F (x) $and some random variables $ Y_1. , , Y_n $ which are independent and identical to CDF distributed $ G (y) $, Suppose that everyone $ X_i $ is independent of everyone $ Y_i $,

These pairs of random variables give random sums. $ X_1 + U_1 $ , , , $ X_n + U_n $, What is the probability that the first random sum is the highest? That's clear $ 1 / n $ through symmetry. Equally clear is the likelihood that the first random sum is the highest Fewer when $ 1 / n $ if I assume that the first random sum falls below a certain limit. This is,

$$ P (X_1 + U_1 geq X_j + U_j hspace {0,1cm} for all i hspace {0,1cm} | hspace {0,1cm} U_i leq u) leq 1 / n $$

Presumably this is true with strict inequality, though $ n> 1 $, Is that true? I've spent a lot of time showing this, and would appreciate any help, ideas, or hints.

Thank you in advance!

algebraic geometry – Is the Zariski topology on a variety $V$ a maximal Noetherian topology?

Let $K$ be an algebraically closed field. By a variety $V$ definable over $K$, I mean a quasi-projective or an algebraic variety in sense of Weil. It is the set of points in an affine or a projective space over some bigger algebraically closed field $L$.

Now consider the Zariski topology $tau$ on $V$ together with Krull dimension on closed sets. Is it possible to enrich this topology in the naive sense by adding new closed sets, so that the enriched topology $tau^prime$ is also a Noetherian topology?

In other words, is the Zariski topology on $V$ a maximal noetherian topology?