2.1. Suppose that F is a finite field with say |F| = pm that V has finite dimension n over F. Then find the order of GL(V). 9 and =
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- i need the answer quickly235.9. (a) Let K/F be an extension of fields. Prove that [K: F] = 1 if and only if K = F. (b) Let L/F be a finite extension of fields, and suppose that [L: F] is prime. Suppose further that K is a field lying between F and L; i.e., F C K C L. Prove that either K = F or K = L.
- 1.4.2 Let F = {a + bi : a, b e Q}, where i? = – 1. Show that F is a field.19.2.8. Let F7 = Z/7Z be the field with seven elements, and let f,g Є F7[x] be defined by f(x) = x³ + x³ + x + 1 and g(x) = x² + x. Is f in the ideal generated by g? If the answer is no, then find h = F7[×] with degree less than two such that f − h is in the ideal generated by g.1. Prove that for any a, b E K in an ordered field K with a < b we have aTheorem 1.2.17 (Intervals) In an ordered field F, the following sets are intervals: (a) [a, b] = {x E F:a ≤x≤b}; (This could be {a} or Ø.) (b) (a, b) = {x E F:a < xExercise 6. Find primitive elenants in (a) the field Z,, (8) the ficeld Z13.Let F be a field of characteristic not equal to 2. Let a, b E F such that b is not a square in E. Show that there exist m, n E F such that Va+√b = √m + √n iff a²-b is a square in F.5.16. Let F be a field, and suppose that the polynomial X² + X + 1 is irreducible in F[X]. Let K = F[X]/(X² + X + 1)F[X] be the quotient ring, so we know from Theorem 5.26 that K is a field. We will put bars over polynomials to indicate that they represent elements of K. In other words, if we let I be the ideal I = (X² + X + 1)F[X], then X + 2 is shorthand for the coset (X + 2) + I. (a) Find a polynomial p(X) € F[X] of degree at most 1 satisfying p(X) = (X+3) · (2X + 1). (b) Find a polynomial q(X) € F[X] of degree at most 1 satisfying q(X) · (X + 1) = I. In other words, find a multiplicative inverse for X + 1 in the field K. (c) Find a polynomial r(X) € F[X] of degree at most 1 satisfying r(x)² = = −3. In other words, find a square root of −3 in the field K.Please solve this question with a good explanation. It is Linear Algebra question.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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