MIT 18.703 Modern Algebra / Assignment 1

[E7-87-83]

Course: MIT 18.703 Modern Algebra
Document: ./MIT/Solutions/18.703/Assignment1/
Date: Early 2023

Assigment 1 - Ex2 all okay。不過 Ex2 ＃12 有typo。

Assigment 1 - Ex4 ＃1-＃5 okay

＃6:

題目：Let ∗ be defined on C by letting a ∗ b = |ab|. Determine whether the binary operation ∗ gives a group structure on the given set.

解： The proof should be:

0 is in C but 0 does not have an inverse with respect to ∗.

Note: The given binary operation ∗ gives a group structure on C \ {0}

The inverse of the equivalent class of (1+i) in C \ {0} respect to * is the equivalent class containing \frac{1-i}{2} , i.e. {e^iθ \frac{1-i}{2} | θ \in R} .

Equivalent class in a group 嘅概念可見於
https://math.berkeley.edu/~gmelvin/math113su14/math113su14notes_online.pdf

Assignment 1 - Ex5 ＃1, ＃3, ＃13, ＃20 okay

＃2: \mathbb{Q}^+ is a not subgroup of the group \mathbb{C} under addition because for any a \in \mathbb{Q}^+, -a \notin \mathbb{Q}^+
or 0 \notin \mathbb{Q}^+ .

Assignment 1 - Ex6 ＃17, ＃19, ＃21, ＃22, ＃28, ＃32, ＃33, ＃34 okay

＃4 無過程！？[雖然好easy...]

Assignment 1 - Bonus ＃1, Bonus ＃2 okay