By Alexander Barg (auth.), Teo Mora, Harold Mattson (eds.)

ISBN-10: 3540631631

ISBN-13: 9783540631637

This booklet constitutes the strictly refereed court cases of the twelfth overseas Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-12, held in Toulouse, France, June 1997.

The 27 revised complete papers provided have been conscientiously chosen through this system committee for inclusion within the quantity. The papers tackle a vast variety of present concerns in coding conception and desktop algebra spanning polynomials, factorization, commutative algebra, genuine geometry, staff conception, and so on. at the mathematical aspect in addition to software program platforms, telecommunication, complexity conception, compression, sign processing, and so on. at the desktop technological know-how and engineering side.

**Read or Download Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 12th International Symposium, AAECC-12 Toulouse, France, June 23–27, 1997 Proceedings PDF**

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**Example text**

Prove that Γ ◦ ι ◦ Γ = Γ, and that ι ◦ Γ ◦ ι = ι. Thus images under either map are always closed. √ 3. Let α = 4 2 ∈ R, and set K = Q(α). Compute the closure of Q in K. 54 CHAPTER 2. FIELD AND GALOIS THEORY 4. 2, let f (x) = x3 + x2 − 2x − 1 ∈ Q[x], and let α ∈ C be a root of f (x). Compute the closure of Q in Q(α). 5. If E ⊇ F is a finite Galois extension, prove that every subgroup of G = Gal(E/F) is closed. 6. Let E ⊇ K ⊇ F with E ⊇ F algebraic. If E is Galois over K and K is Galois over F, must it be true that E is Galois over F?

5 Let F ⊆ K be an algebraic extension of fields where F has characteristic p > 0. If α ∈ K, then α is separable over F if and only if F(α) = F(αp ). 6 Let F ⊆ K be an algebraic extension, where F is a field of characteristic p > 0. Let α ∈ K be an inseparable element over F. The following are equivalent: (i) α is purely inseparable over F. e (ii) The minimal polynomial has the form mα (x) = xp − a ∈ F[x], for some positive integer e and for some a ∈ F. , α. Let F be a field of characteristic p > 0.

B) 2 + ζ, where ζ = e2πi/3 . 2. Let F ⊆ K be a field extension with [K : F] odd. If α ∈ K, prove that F(α2 ) = F(α). 3. Assume that α = a + bi ∈ C is algebraic over Q, where a is rational and b is real. Prove that mα (x) has even degree. √ √ 4. Let K = Q( 3 2, 2) ⊆ C. Compute [K : Q]. √ 5. Let K = Q( 4 2, i) ⊆ C. Show that (a) K contains all roots of x4 − 2 ∈ Q[x]. (b) Compute [K : Q]. 46 CHAPTER 2. FIELD AND GALOIS THEORY 6. Let F = C(x), where C is the complex number field and x is an indeterminate.

### Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 12th International Symposium, AAECC-12 Toulouse, France, June 23–27, 1997 Proceedings by Alexander Barg (auth.), Teo Mora, Harold Mattson (eds.)

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