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--- math121a-f23:september_6_wednesday [2023/09/05 11:12]
pzhou
+++ math121a-f23:september_6_wednesday [2023/09/07 01:10] (current)
pzhou [post lecture note]
@@ Line 15: / Line 15: @@
 (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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