Hovhannes Khudaverdian's notes

... For example σ12 (w + w0 ) = w0 + w. We come to very important conclusion: The expression w(x1 .x2 .x3 ) = x21 x2 + x22 x3 + x23 x1 for roots of the cubic equation is a root of a quadratic equation with rational coefficients. We can calculate it by solving the equation of lower degree. ...

... For example σ12 (w + w0 ) = w0 + w. We come to very important conclusion: The expression w(x1 .x2 .x3 ) = x21 x2 + x22 x3 + x23 x1 for roots of the cubic equation is a root of a quadratic equation with rational coefficients. We can calculate it by solving the equation of lower degree. ...

On different notions of tameness in arithmetic geometry

... As p - v R (b0 ), this implies p|e and, as e ≤ p, this indeed means e = p. Proof of the Key Lemma 2.4. Let Y be the normalization of X in k(Y 0 ). Then Y = Spec( B) for some semi-local ring B. Reduction to dim( A) = 2. The assertion is trivial if dim( A) = 1. Assume that we have shown Lemma 2.4 for ...

... As p - v R (b0 ), this implies p|e and, as e ≤ p, this indeed means e = p. Proof of the Key Lemma 2.4. Let Y be the normalization of X in k(Y 0 ). Then Y = Spec( B) for some semi-local ring B. Reduction to dim( A) = 2. The assertion is trivial if dim( A) = 1. Assume that we have shown Lemma 2.4 for ...

Topological realizations of absolute Galois groups

... For algebraically closed fields F , the space XFM would have to be constructed in such a way as to freely adjoin the Steinberg relation on its cohomology groups; the general case should reduce to this case by descent. Descent along the cyclotomic extension. So far, all of our results were assuming t ...

... For algebraically closed fields F , the space XFM would have to be constructed in such a way as to freely adjoin the Steinberg relation on its cohomology groups; the general case should reduce to this case by descent. Descent along the cyclotomic extension. So far, all of our results were assuming t ...

A refinement of the Artin conductor and the base change conductor

... potentially ordinary reduction; in some sense, we are using analytic K-tori as a link to connect A with an algebraic K-torus. In [Cha00, § 6], the existence of such a formula was conjectured based on the fact that c(A) is an isogeny invariant if A has potentially ordinary reduction; see [Cha00, Theo ...

... potentially ordinary reduction; in some sense, we are using analytic K-tori as a link to connect A with an algebraic K-torus. In [Cha00, § 6], the existence of such a formula was conjectured based on the fact that c(A) is an isogeny invariant if A has potentially ordinary reduction; see [Cha00, Theo ...

On the sum of two algebraic numbers

... PAULIUS DRUNGILAS, ARTŪRAS DUBICKAS, CHRIS SMYTH Abstract. For all but one positive integer triplet (a, b, c) with a 6 b 6 c and b 6 6, we decide whether there are algebraic numbers α, β and γ of degrees a, b and c, respectively, such that α + β + γ = 0. The undecided case (6, 6, 8) will be include ...

... PAULIUS DRUNGILAS, ARTŪRAS DUBICKAS, CHRIS SMYTH Abstract. For all but one positive integer triplet (a, b, c) with a 6 b 6 c and b 6 6, we decide whether there are algebraic numbers α, β and γ of degrees a, b and c, respectively, such that α + β + γ = 0. The undecided case (6, 6, 8) will be include ...

Miles Reid's notes

... As we see many times later in the course, finding one root α of a polynomial f (x) in general only allows us to pull one factor x − α out of f , giving f (x) = (x − α)g(x). It certainly happens sometimes that g(x) is irreducible of degree n − 1 (see Example 3.23), and we have more work to do to fin ...

... As we see many times later in the course, finding one root α of a polynomial f (x) in general only allows us to pull one factor x − α out of f , giving f (x) = (x − α)g(x). It certainly happens sometimes that g(x) is irreducible of degree n − 1 (see Example 3.23), and we have more work to do to fin ...

1. Outline of Talk 1 2. The Kummer Exact Sequence 2 3

... 3. µn : Define µn over Z as Spec Z[x]/(x − 1). The corresponding etale sheaf over X sends U to the group of nth roots of unity in Γ(U, OU ). Over an algebrically closed field of characteristic not dividing n, this is the constant sheaf Z/nZ. The second and third example actually give us an example o ...

... 3. µn : Define µn over Z as Spec Z[x]/(x − 1). The corresponding etale sheaf over X sends U to the group of nth roots of unity in Γ(U, OU ). Over an algebrically closed field of characteristic not dividing n, this is the constant sheaf Z/nZ. The second and third example actually give us an example o ...

HHG-published (pdf, 416 KiB) - Infoscience

... “correct” definition of homotopic Hopf–Galois extensions may require categorification of condition (1) of Definition 0.1 as well. For the purposes of this paper, we have chosen not to do so, but further experience with this notion may lead to the consensus that one should. The key problem that we mu ...

... “correct” definition of homotopic Hopf–Galois extensions may require categorification of condition (1) of Definition 0.1 as well. For the purposes of this paper, we have chosen not to do so, but further experience with this notion may lead to the consensus that one should. The key problem that we mu ...

ARIZONA WINTER SCHOOL 2014 COURSE NOTES

... In particular, if ℘i are the places of K dividing p, then Kp = i K℘i . So in particular, we see that the algebra Kp contains the information of the splitting/ramification type of p. The n homomorphisms K → Q̄ extend to n homomorphisms Kp → Q̄p , and we have a map then from the decomposition group Ga ...

... In particular, if ℘i are the places of K dividing p, then Kp = i K℘i . So in particular, we see that the algebra Kp contains the information of the splitting/ramification type of p. The n homomorphisms K → Q̄ extend to n homomorphisms Kp → Q̄p , and we have a map then from the decomposition group Ga ...

The Essential Dimension of Finite Group Schemes Corso di Laurea Magistrale in Matematica

... definition for L/K. In fact consider our previous example of k( t)/k(t). Take as Kn = k(t2n+1 ) and Ln = k(tn+1/2T ). Then L/K is surely defined over Kn for each n but not on their intersection n Kn = k. This thesis will not try to follow this direct approach, which proves itself inconvenient when t ...

... definition for L/K. In fact consider our previous example of k( t)/k(t). Take as Kn = k(t2n+1 ) and Ln = k(tn+1/2T ). Then L/K is surely defined over Kn for each n but not on their intersection n Kn = k. This thesis will not try to follow this direct approach, which proves itself inconvenient when t ...

FILTERED MODULES WITH COEFFICIENTS 1. Introduction Let E

... • its Galois type, i.e., the Weil-Deligne representation W D(ρ)|Ip , and, • its Hodge-Tate weights. The following question is of interest: given a Galois type τ of degree 2 and an integer k > 1, what are the (isomorphism classes of) potentially semi-stable representations ρ : Gp → GL2 (E) having typ ...

... • its Galois type, i.e., the Weil-Deligne representation W D(ρ)|Ip , and, • its Hodge-Tate weights. The following question is of interest: given a Galois type τ of degree 2 and an integer k > 1, what are the (isomorphism classes of) potentially semi-stable representations ρ : Gp → GL2 (E) having typ ...

roots of unity - Stanford University

... Lagrange resolvant enters, but what Lagrange resolvant? Often Hilbert’s Theorem 90 is used, or linear independence of characters, as in my Paragraph 26. But perhaps this is a disservice to students. It is not all that hard to exhibit explicit non-zero Lagrange resolvants in the presence of primitive ...

... Lagrange resolvant enters, but what Lagrange resolvant? Often Hilbert’s Theorem 90 is used, or linear independence of characters, as in my Paragraph 26. But perhaps this is a disservice to students. It is not all that hard to exhibit explicit non-zero Lagrange resolvants in the presence of primitive ...

An explicit example of a noncrossed product division algebra

... In general, it is very difficult to explicitly compute outer automorphisms of division algebras or central simple algebras. But in very special cases the problem can be reduced to the solution of norm equations in field extensions (see § 3). Since we work over number fields, methods from computation ...

... In general, it is very difficult to explicitly compute outer automorphisms of division algebras or central simple algebras. But in very special cases the problem can be reduced to the solution of norm equations in field extensions (see § 3). Since we work over number fields, methods from computation ...

Galois Groups and Fundamental Groups

... (In this case, the universal cover is analogous to the separable algebraic closure of the field.) One subgroup is contained in another iff one cover dominates the other. The whole group corresponds to X, and the trivial subgroup corresponds to its universal cover X̃. There is a notion of degree of ...

... (In this case, the universal cover is analogous to the separable algebraic closure of the field.) One subgroup is contained in another iff one cover dominates the other. The whole group corresponds to X, and the trivial subgroup corresponds to its universal cover X̃. There is a notion of degree of ...

# Évariste Galois

Évariste Galois (French: [evaʁist ɡaˈlwa]; 25 October 1811 – 31 May 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel.