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102年 - 102 淡江大學 轉學考 代數#53069
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2. (12 pts) Prove or disprove: If G is a group of order 53, then G must be cyclic.
其他申論題
5. Find the Huffman tree for the following characters using Huffman coding with the given frequencies: A(12), B(8), C(9), D(20), E(31), F(14), G(8). (15%)
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6. Multiple Choice Questions (10%)
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【已刪除】 (a) Suppose p is a prime number and a is an integer, (a,p) — 1. Prove that ap~11 (mod p).
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(b) What is the remainder when 3535 is divided by 37?
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3. (12 pts) Suppose G = {e, a, 6, c} is a group of order 4; but it contains no element of order 4. Write out the operation table for G.
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(a) Prove that every finite integral domain is a field.
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(b) Give an example of an integral domain which is not a field.
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5. (12 pts) Show that the principal ideal (x — 1) in Z[x] is prime but not maximal. .
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(a) Show that x3 + x + 1 is irreducible in Z5 [x].
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(b) Let R be the quotient ring Z5[x]/ (x3 +x + 1). How many elements are there in R1 Is R a field? Please justify your answer
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