Math 361, Spring 2018, Assignment 9
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Read:[edit]
- Section 30.
- Section 31.
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Vector space (over an arbitrary field F).
- Subspace.
- Homomorphism (between two vector spaces; a.k.a. linear transformation).
- Monomorphism.
- Epimorphism.
- Isomorphism.
- Image (of a linear transformation).
- Kernel (of a linear transformation).
- Linear combination (of elements of some set S in some vector space V).
- Span (of some set S in some vector space V).
- Linear relation (among elements of S).
- Trivial linear relation.
- Linearly independent set.
- Basis.
- Dimension.
- [E:F].
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem concerning the existence of bases.
- Theorem concerning extension of linearly independent sets to bases.
- Theorem concerning refinement of spanning sets to bases.
- Theorem concerning cardinalities of two bases for the same space.
- Formula for [F(β):F] when β is algebraic over F.
- Formula for [F(β):F] when β is transcendental over F.
- Theorem concerning transcendental elements of finite-dimensional extensions.
- Theorem constraining the order of a finite field.
- Theorem concerning the existence of fields of order pd (we did not prove this yet, but we stated it).
- Theorem relating finite fields with equal cardinalities (we did not prove this yet).
- Theorem concerning the number of elements required to generate a finite field over Zp (we did not prove this yet).
- Corollary concerning the degrees of irreducible polynomials in Zp[x].
- Dimension formula (relating [E:F] to [E:K] and [K:F]).
Solve the following problems:[edit]
- Section 30, problems 1, 5, 7, 8, and 9.
- Section 31, problems 1, 3, 5, and 7.
- Construct the field with 125 elements. You do not need to make full addition and multiplication tables; instead, write down two fairly typical elements of your field, and show how to add and multiply them. (Hint: you do not need to run the Sieve of Eratosthenes. How else can you tell whether a cubic polynomial is irreducible?)
- Suppose that F→E is any finite-dimensional extension, and suppose that K is any subextension. Prove that [K:F] is a divisor of [E:F].
- Prove that √2∉Q(3√2). (Hint: if it were, then Q(√2) would be a subextension of Q(3√2).)