1. What is the explicit isomorphism problem over a field?
The explicit isomorphism problem over a field involves constructing an explicit isomorphism for a given F-algebra A assumed isomorphic to M^d (F) for some positive integer d. It is a natural problem in computational representation theory, aiming to compute the Jacobson radical of A, the decomposition of the semi-simple part of A as a sum of simple F-algebras, and finding an isomorphism A^n (D) M^m (K) [15, 26, 33]. Applications include arithmetic geometry, algebraic complexity theory, and error-correcting codes. Novel algorithms present a central simple F-algebra A using Brauer factor sets, computed from a generic element with integer coefficients. The probability of irreducible characteristic polynomial for random matrices is 1 - o(1) [17].
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2. What algorithm was introduced for finite base field?
In the case of a finite base field, a polynomial-time algorithm was introduced by Ronyai in [40]. This algorithm is used for solving instances of the problem for Q-algebras isomorphic to M n (Q).
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3. What are Brauer factor sets and their computational challenges?
Brauer factor sets are based on a degree deg A extension K of the base field F and take values in the Galois closure of K. While known algebraically since 1928, there are non-trivial obstacles to using them computationally. Our algorithms apply when Gal(K/F) = Sn and avoid computations in the Galois closure of K. This representation yields polynomial-time classical algorithms. An advantage over presentations as cyclic or crossed-product algebras is that it is easy to find a Brauer factor set representing an arbitrary algebra. The computational challenges lie in efficiently finding and utilizing these factor sets for computations.
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4. What are Brauer factor sets and their properties?
Brauer factor sets are defined and studied in Section 2 of the paper. They are relevant known results that provide a foundation for the research. These sets have specific definitions and properties that are crucial for understanding the behavior of central simple algebras in the context of Galois groups. By recalling these definitions and properties, the paper establishes a solid groundwork for further analysis and exploration of the subject matter.
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5. What is the centraliser of x in A?
The centraliser of x in A, denoted as C A (x), is the set of elements y in A such that xy = yx. It represents the elements that commute with x in the K-algebra A. This concept is important in the study of algebraic structures and their properties. The centraliser plays a crucial role in understanding the symmetry and structure of algebraic systems, as it helps identify elements that preserve certain properties or operations within the algebra. By studying the centraliser, researchers can gain insights into the behavior and relationships of elements within the algebra, which can be useful in various applications, such as solving equations, analyzing symmetries, and exploring algebraic structures.
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6. What is the significance of knowing a cyclic or crossed-product presentation of a central simple F-algebra?
Knowing a cyclic or crossed-product presentation of a central simple F-algebra is significant because it is computationally equivalent to knowing an embedding K-A, where K/F is a cyclic or Galois extension of degree d. This equivalence allows for the computation of groups of units in F and K, which is crucial for understanding the structure and properties of the algebra. However, it is important to note that there is currently no known efficient algorithm for computing such an embedding from the structure constants of an algebra A M d (F). This highlights the complexity and challenges involved in working with central simple F-algebras and emphasizes the importance of further research in this area.
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7. How does knowing a zero divisor of rank larger than 1 help in computing an explicit isomorphism?
Knowing a zero divisor z A of rank larger than 1 allows us to reduce the problem to computing an isomorphism A ' M d ' (F), with d' < d. By using linear algebra, the left ideal L z admits a right unit e, which is an idempotent of rank r < d. This idempotent e can be used to compute an idempotent of rank 1 in A ' and A. By solving the explicit isomorphism problem for A ', we can find an idempotent of rank 1 in A, which also has rank 1 in A. This approach, as mentioned in [33], helps in solving the explicit isomorphism problem efficiently.
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8. What is the classical algorithm for computing discrete logarithms in unit groups?
The classical algorithm for computing discrete logarithms in unit groups is Algorithm 5.3.10. It runs in polynomial time and uses the logarithmic embedding into R n to reduce a multiplicative problem to an additive one. The image of this embedding is a lattice, and the target unit is represented as a linear combination of lattice vectors. The solution is rounded up to the nearest integer vector, and if necessary, the precision can be increased. Alternatively, the problem can be viewed as a bounded distance decoding problem using Babai's algorithm. If the unit has finite order, it can be written as a power of a primitive root of unity. For units with infinite order, generators of the torsion-free part of the unit group are used to define a homomorphism from Z k+1 to the unit group. Generators for the kernel of this homomorphism can be computed in quantum polynomial time, and finding an element in the kernel with the last coordinate as 1 completes the computation.
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9. What are the three algorithmic problems related to class groups and groups of S-Units in number fields?
The three algorithmic problems related to class groups and groups of S-Units in number fields are: (i) Compute generators of Cl(L), (ii) Compute generators of U L, and (iii) Given a principal ideal I, find a generator element of I. These problems are important in number theory and have been the focus of research in developing polynomial-time quantum algorithms. Class groups and unit groups are significant objects in number theory, and their computation is crucial for various applications in the field.
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10. What are Brauer factor sets and central simple algebras?
Brauer factor sets and central simple algebras are mathematical concepts used in the study of algebraic structures. Brauer factor sets are maps that satisfy specific homogeneity and cocycle conditions, and they play a role similar to Galois 2-cocycles when K is not a Galois extension of F. Central simple algebras, denoted as B(K, c), are constructed using homogeneous matrices of K/F and a twist of matrix multiplication using the factor set c. These algebras have an F-algebra structure and can be explicitly computed using methods from section 2.1.2. The space of homogeneous matrices of K/F, denoted as B(K, c), is denoted as M d (F), and the isomorphism between B(K, 1) and M d (F) can be computed explicitly.
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11. How do factor sets yield isomorphic algebras?
Factor sets yield isomorphic algebras by characterizing them through an equivalence relation. In the given section, a homomorphism is defined from the group of reduced homogeneous matrices to the group of reduced Brauer factor sets. This homomorphism, denoted as (m) = (c ijk ) i,j,k[d], where c ijk = m ij m jk m -1 ik, establishes an equivalence relation between factor sets. Two factor sets, c and c', are considered associated if there exists a homogeneous matrix m such that c' = c(m). This association characterizes factor sets that produce isomorphic algebras. Additionally, if c is a reduced factor set associated with the constant factor set 1, it is called trivial, and a reduced homogeneous matrix m exists such that c = (m). The proposition further states that if c and c' are associated factor sets, with c' = c(m), then the map B(K, c) - B(K, c') (l ij) - (l ij m ij -1) is an isomorphism of F-algebras.
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12. How to compute a factor set c for K/F in a central simple F-algebra?
To compute a factor set c for K/F in a central simple F-algebra, follow these steps: (i) Find a vector vA such that A = KvK. (ii) Establish an isomorphism A K [d] M d (K [d] ) mapping th 1 to Diag(th 1, ..., th d). (iii) Identify A K [d] with M d (K [d] ) and treat elements of A as matrices. (iv) Define c ikj := v ik v kj v-1 ij. (v) Use the map (l ij v ij ) - (l ij ) to create an isomorphism A - B(K, c).
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13. What is k-transitive permutation group?
A k-transitive permutation group is a subgroup of the symmetric group S n acting on a set X of cardinal n. For 1 <= k <= n, the group G is k-transitive if for all tuples (x 1 , . . ., x k ) and (y 1 , . . ., y k ) X k, where both x i and y i are pairwise distinct, there exists an element s in H such that s(x i ) = y i for all i [k].
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14. How does a factor set determine values?
In a 3-transitive Galois group setting, a factor set is entirely determined by one of its values. This is proven by showing that c i'j'k' equals sc ijk, and c iij equals c ijj, c ' iij equals c ' ijj, and c iji equals c ' iji for i = j. The remaining proof involves applying Equation 2 to k [d] \ {i, j}, resulting in c iji = c ijk c jik, which equals c ' iji. This demonstrates that a factor set's values uniquely determine its properties.
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15. What is the purpose of Proposition 6 in the context of factor sets and multiplication in B(K, c)?
Proposition 6 establishes a formula for multiplying elements in B(K, c) using isomorphism ph. It provides a way to compute the product of two elements (a, b) and (a', b') in B(K, c) by using the isomorphism ph to represent these elements as tuples in K x K [2]. The formula states that the product of (a, b) and (a', b') is equal to (a''', b'''), where a''' and b''' are calculated using the given formulas. This allows for efficient multiplication in B(K, c) by leveraging the isomorphism ph and the underlying vector space isomorphism between B(K, c) and K x K [2]. The purpose of Proposition 6 is to provide a compact representation of factor sets, their trivialisations, and elements of B(K, c), enabling polynomial-time computation of arithmetical operations in B(K, c).
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16. What is the polynomial-time algorithm for computing a factor set from an embedding in a central simple algebra?
There exists a polynomial-time algorithm which takes as input a positive integer d, structure constants for a central simple F -algebra A of degree d and coordinates for some u A such that F (u)/F is a degree d separable field extension and Gal(F (u)/F ) is 3-transitive, and outputs a reduced factor set c for F (u)/F , together with an explicit isomorphism ph : A - B(K, c). The algorithm involves computing an isomorphism A ~ = B(K, c) for a central simple algebra A given with an embedding K A. It follows the construction given in Section 2.2.3. The algorithm computes some v A such that A = KvK. It begins with a lemma and a proposition, and uses linear algebra and the properties of the algebra to find coordinates for v A. The map ps : A - B(K, c) is computed, and the factor set c is computed to make this map an F -algebra homomorphism. The algorithm avoids direct computation in K [d] and uses the separable extension property of K [3] /K [2] to ensure the uniqueness of the solution. The algorithm is efficient and polynomial-time, making it suitable for computing factor sets from embeddings in central simple algebras.
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17. What is the generalization of Hilbert's theorem 90 to ideals and Brauer factor sets?
The generalization of Hilbert's theorem 90 to ideals and Brauer factor sets is Lemma 5. It states that if I is a fractional ideal in K [2] with support disjoint from S, such that (I) = Z K [3], then there exists a fractional ideal J of K [1] such that I = Ji 2 (J) -1. This generalization extends Hilbert's theorem 90 by applying it to ideals instead of elements and by considering Brauer factor sets instead of Galois cocycles. The proof of Lemma 5 is a generalization of the proof of [20, Lemma 9], which generalized Hilbert's theorem 90 to ideals. The proof involves setting J := d i=2 i 1,i (I) and showing that I = Ji 2 (J) -1. This generalization is a significant advancement in the field of number theory and algebraic geometry.
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18. How does Algorithm 1 compute fractional ideals for a trivial Brauer factor set?
Algorithm 1 computes fractional ideals a1, ..., an of K representing the generators of the class group of K. It follows a series of steps: 1. Compute fractional ideals a1, ..., an. 2. Remove ideals pMK where vp(ai) = 0. 3. Remove ideals pMK where p divides the discriminant of K. 4. Remove ideals pMK where vp(c) = 0. 5. Compute generators g1, ..., gr of UK[2], S. 6. Compute generators h1, ..., hs of UK[3], S. 7. Compute A = (aij)MSr(Z) such that (gj) = si=1 hai,j. 8. Compute (b1, ..., bs) Zs such that c = si=1 hbi i. 9. Compute C = (c1, ..., cr) Zr such that AC = (b1, ..., bs). Finally, return ri=1 gci i 5. This process helps in trivializing a Brauer factor set for K/F.
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19. What is the probability of success for irreducible characteristic polynomial?
The probability of success for an irreducible characteristic polynomial is at least 1/u(d) for some polynomial u. This lower bound is sufficient for the needs of the work, although it is not sharp. The probability of success is not required to go to 1 as the parameters grow. Proposition 10 states that for a random matrix x as in Heuristic 1, the probability of Gal(khx) = Sn or An with probability 1 - o(1) is bounded below uniformly with respect to the number field F. The proof of Proposition 10 involves Lemma 6, which describes conjugacy classes that may not be contained together in a proper transitive subgroup of Sn. The proof also utilizes Lemma 7, which states that if G is a transitive subgroup of Sn, and G contains a n-1-cycle and either a permutation of cycle structure (2, n-3, 1) if n is even, or (2, n-2) if n is odd, then G = Sn. This information is crucial in understanding the probability of success for irreducible characteristic polynomials in matrix algebras.
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20. How can we prove uniform random variable in M d (k 2 ) x M d (k 3 )?
To prove that our sampling method produces a uniform random variable in M d (k 2 ) x M d (k 3 ), we need to show that if x is a uniform random variable in i[n] [6]a i, then (p 2 (x), p 3 (x)) is a uniform random variable in M d (k 2 ) x M d (k 3 ). The proof involves demonstrating that the set i[n] [6]a i is x = F 1 F 2, where F 1 and F 2 are monic irreducible polynomials of degrees 1 and d-1 respectively. The probability that there exist such polynomials F 1 and F 2, and kh x = F 1 F 2 is denoted as p'. By using the Mobius function and the properties of the number of monic irreducible polynomials, we can show that p' is bounded away from zero. Additionally, we prove that F (q, d) is bounded away from zero, and that the probabilities for specific factorizations of kh x are bounded from below. By combining these results, we can conclude that with probability greater than 1/u(d), a uniform x in i[n] [6]a i has the desired property. This demonstrates that our sampling method produces a uniform random variable in M d (k 2 ) x M d (k 3).
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21. What is the significance of d=2 case?
The d=2 case is significant as it represents a scenario where every A is isomorphic to a quaternion algebra over F. This case is included for completeness, even though it is a known result. The main idea involves finding a trace 0 element uA and an element v such that uv = -vu by solving a system of linear equations. This leads to a quaternion basis. If given a quaternion algebra with the presentation u^2 = a, v^2 = b, finding a zero divisor is equivalent to solving the norm equation NF(a|F) = b. By invoking results from Section 2.1.5, an explicit isomorphism between A and M2(F) can be found in quantum polynomial time. The proof involves computing a basis for a maximal Z-order of A, using Heuristic 1 and Proposition 10 to find a polynomial u, and applying Theorem 2 to compute a reduced Brauer factor set cK and an isomorphism ph. Finally, a trivialisation m of c is found using Algorithm 1, and a rank one zero divisor of B(K, c) is computed using Corollary 7. This process leads to the deduction of the isomorphism ps as explained in Section 2.1.2.
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22. How can Algorithm 2 be modified for efficiency?
To improve the efficiency of Algorithm 2, we can modify it by handling cases where kh x is not irreducible. After line 2, we test if the minimal polynomial P x is reducible or a proper divisor of kh x. If P x is reducible, we compute eAe, a subalgebra isomorphic to M r (F), and apply the algorithm recursively to A '. If P x is a proper divisor, we apply the algorithm recursively to C A (x) with F (x) as a new base field. This unconditional version of Algorithm 2 runs in polynomial time if d is bounded. It succeeds with probability larger than f rac1u(d), where u is a polynomial and d is the degree of the input algebra. The modified algorithm uses a bounded amount of recursive calls and performs computations in polynomial time in both the new base field and the new input algebra. The size of the structure constants over the new base field and the new input algebra is polynomial in the parameters, ensuring efficient computation.
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