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5.8. Consider the set F4 {0, 1, a, B} consisting of 4 elements. Define an addition law and a multiplication law on F4 using Figure 15. (a) Prove that these rules make F4 into a field. (It's quite tedious to check the associative law directly by writing out all triple products, so either try to find a clever way to check it or just verify a few instances; e.g., (aß)a = a(Ba).)

5.8. Consider the set F4 {0, 1, a, B} consisting of 4 elements. Define an addition law and a multiplication law on F4 using Figure 15. (a) Prove that these rules make F4 into a field. (It's quite tedious to check the associative law directly by writing out all triple products, so either try to find a clever way to check it or just verify a few instances; e.g., (aß)a = a(Ba).)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**5.8.** Consider the set \( \mathbb{F}_4 = \{0, 1, \alpha, \beta\} \) consisting of 4 elements. Define an addition law and a multiplication law on \( \mathbb{F}_4 \) using Figure 15. (a) Prove that these rules make \( \mathbb{F}_4 \) into a field. (It's quite tedious to check the associative law directly by writing out all triple products, so either try to find a clever way to check it or just verify a few instances; e.g., \( (\alpha \beta) \alpha = \alpha (\beta \alpha) \).)
Transcribed Image Text:**5.8.** Consider the set \( \mathbb{F}_4 = \{0, 1, \alpha, \beta\} \) consisting of 4 elements. Define an addition law and a multiplication law on \( \mathbb{F}_4 \) using Figure 15. (a) Prove that these rules make \( \mathbb{F}_4 \) into a field. (It's quite tedious to check the associative law directly by writing out all triple products, so either try to find a clever way to check it or just verify a few instances; e.g., \( (\alpha \beta) \alpha = \alpha (\beta \alpha) \).)
**Figure 15. Addition and Multiplication Tables for F<sub>4</sub>** The image shows two tables representing the addition and multiplication operations in the finite field F<sub>4</sub>. **Addition Table:** - The table headers show columns: `0`, `1`, `α`, `β`. - Rows represent the addition result for the values: `0`, `1`, `α`, `β`. | + | 0 | 1 | α | β | |----|---|---|---|---| | 0 | 0 | 1 | α | β | | 1 | 1 | 0 | β | α | | α | α | β | 0 | 1 | | β | β | α | 1 | 0 | **Multiplication Table:** - Columns and rows are labeled with `0`, `1`, `α`, `β`. - Each cell shows the multiplication result for the corresponding row and column values. | × | 0 | 1 | α | β | |----|---|---|---|---| | 0 | 0 | 0 | 0 | 0 | | 1 | 0 | 1 | α | β | | α | 0 | α | β | 1 | | β | 0 | β | 1 | α | These tables summarize how elements in F<sub>4</sub> combine under addition and multiplication, which are crucial for understanding operations in this field.
Transcribed Image Text:**Figure 15. Addition and Multiplication Tables for F<sub>4</sub>** The image shows two tables representing the addition and multiplication operations in the finite field F<sub>4</sub>. **Addition Table:** - The table headers show columns: `0`, `1`, `α`, `β`. - Rows represent the addition result for the values: `0`, `1`, `α`, `β`. | + | 0 | 1 | α | β | |----|---|---|---|---| | 0 | 0 | 1 | α | β | | 1 | 1 | 0 | β | α | | α | α | β | 0 | 1 | | β | β | α | 1 | 0 | **Multiplication Table:** - Columns and rows are labeled with `0`, `1`, `α`, `β`. - Each cell shows the multiplication result for the corresponding row and column values. | × | 0 | 1 | α | β | |----|---|---|---|---| | 0 | 0 | 0 | 0 | 0 | | 1 | 0 | 1 | α | β | | α | 0 | α | β | 1 | | β | 0 | β | 1 | α | These tables summarize how elements in F<sub>4</sub> combine under addition and multiplication, which are crucial for understanding operations in this field.
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