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--- math121a-f23:september_6_wednesday [2023/09/05 18:12]
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+++ math121a-f23:september_6_wednesday [2026/02/21 14:41] (current)
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-(optional) 5. other weird "numbers"? (if you walk like a number, talk like a number, operate like a number, I will call you a number!) The notion of a 'field'.
+(optional) other weird "numbers"? (if you walk like a number, talk like a number, operate like a number, I will call you a number!) The notion of a 'field'.
+===== post lecture note =====
+Roughly speaking, a 'ring' is a set whose elements can do addition and multiplication (among) themselves. Example: $\Z$, polynomial. (to be precise, we talk about commutative multiplication, that satisfies $x \cdot y=y\cdot x$. matrix multiplication may not be commutative.)
+A 'field', is a ring where any nonzero element has a multiplicative inverse. $\Z$ is not a field. $\Q$, $\R$, $\C$ are field.
+$\gdef\F{\mathbb F}$
+We talked about finite field. Given a prime number $p$, we define $\F_p = \Z / p\Z$, This notation may reminds you of the quotient vector space $V/W$, indeed, $\Z / p \Z$ is the set of equivalence class, where we say two integers $n_1, n_2$ are equivalent (and write $n_1 \equiv n_2 (mod p)$), if $n_1 - n_2 \in p \Z$, i.e. the difference is a multiple of $p$. In class, we set $p=7$, and we say $1 \equiv 8 (mod 7)$. If we use $[n] = n + p \Z$ the equivalence class that $n$ belongs to, then we write $[1]=[8]$.
+$$\F_7 = \{ [0], [1], \cdots, [6]\}$$
+We have arithematics like
+$$ [a] + [b] = [a+b], \quad [a] \cdot [b] = [ab]. $$
+For example, $[2] \cdot [4] = [8] = [1]$. (When there is no danger of confusion, we just write $n$ for $[n]$)
+Question (optional): \\
+. Can you write down how $\F_5$ behave? For example, what is $[2] + [4] = ?$ What is who multiply $[3]$ equasl $[1]$?
+. 'finite field version of complex number'. Take $K$ be a field. We consider $K[\sqrt{-1}] := K[x] / (x^2+1) = \{a + b x \mid a, b \in K, x^2 = -1\}$. Is this always a field? Namely, can you always define $1/(a+bx)$? We see that for $K = \R$, this works, $1/(a+bx) = (a-bx) / (a^2 + b^2)$. Does the inverse always exist for $K[\sqrt{-1}]$? Try $K = \Q,  \F_5,  \F_7$, see if you can find some pattern.
+. In class, we also talked about, can you define 'super complex number', that instead of using two real numbers $a,b$ to represent a complex number $a + b i$, but three real numbers? For example, we can try
+$$ \R[x]/(x^3-1) = \{ a+b x + c x^2 \mid a,b,c \in \R,  x^3=1 \} $$
+can you define multiplication on it? $ ( 1+ x + x^2) (2 + x) = ?$
+Does every nonzero element has a (multiplicative) inverse? For example, $x$ has inverse,
+$$ 1/x = x^2. $$
+$x+1$ has inverse, we have
+$$ \frac{1}{1+x} = \frac{1-x+x^2}{(1+x)(1-x+x^2)} = \frac{1-x+x^2}{1+x^3} = \frac{1-x+x^2}{2}. $$
+Does $x-1$ has inverse?
+-------
+you may ask:  why we care about other 'field'? I am happy with $\C$ and $\R$. I don't have a good answer for that, maybe you will find some application some day.
+==== Exercise ====
+(part of homework)
+Read Boas Ch2, section 1 - 9,  find 5 interesting problems there and do it. (copy down the problem, so the grader / reader know which one you are doing).

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