1. ## Linear Algebra|Fields

1) can anyone explain me about characteristic of fields? is in 1+1+1+1+1
+ defined as the addition operator of F? what about a*n when a belongs to F and n is char F.i know it eqauls 0 by making that a=a*1 and n*1 is 0 but is the * the multiplying operator of F because n doesn't have to belong to F.
2)Let F be a field of characteristic p (prime number). prove that Zp is contained in F. Zp={0,1,.....,p-1} regarding modulo p.
i know that 1 belongs to F as weel as 0. but is 1+1=2 in F ?
Thank You Guys.

2. Originally Posted by Tom1234
1) can anyone explain me about characteristic of fields? is in 1+1+1+1+1
+ defined as the addition operator of F? what about a*n when a belongs to F and n is char F.i know it eqauls 0 by making that a=a*1 and n*1 is 0 but is the * the multiplying operator of F because n doesn't have to belong to F.
in F we define $\displaystyle n=n \cdot 1_F=\underbrace {1_F + \cdots + 1_F}_{\rm n \ times},$ if $\displaystyle n > 0$ and $\displaystyle n=n \cdot 1_F=-\underbrace {(1_F + \cdots + 1_F)}_{\rm -n \ times},$ if $\displaystyle n < 0.$

2)Let F be a field of characteristic p (prime number). prove that Zp is contained in F. Zp={0,1,.....,p-1} regarding modulo p.
i know that 1 belongs to F as weel as 0. but is 1+1=2 in F ?
Thank You Guys.
define $\displaystyle f: \mathbb{Z} \longrightarrow F$ by $\displaystyle f(n)=n \cdot 1_F.$ see that $\displaystyle f$ is a ring homomorphism. also $\displaystyle n \cdot 1_F=0$ if and only if $\displaystyle p \mid n.$ so $\displaystyle \ker f = p \mathbb{Z}.$ thus $\displaystyle \mathbb{Z}/p \mathbb{Z}$ is isomorphic to a subring of $\displaystyle F.$

3. Originally Posted by NonCommAlg
in F we define $\displaystyle n=n \cdot 1_F=\underbrace {1_F + \cdots + 1_F}_{\rm n \ times},$ if $\displaystyle n > 0$ and $\displaystyle n=n \cdot 1_F=-\underbrace {(1_F + \cdots + 1_F)}_{\rm -n \ times},$ if $\displaystyle n < 0.$

define $\displaystyle f: \mathbb{Z} \longrightarrow F$ by $\displaystyle f(n)=n \cdot 1_F.$ see that $\displaystyle f$ is a ring homomorphism. also $\displaystyle n \cdot 1_F=0$ if and only if $\displaystyle p \mid n.$ so $\displaystyle \ker f = p \mathbb{Z}.$ thus $\displaystyle \mathbb{Z}/p \mathbb{Z}$ is isomorphic to a subring of $\displaystyle F.$
Thank you but actually i don't know what isompophic means or ring.
All i know is Fields and Characteristic while i know the rules for a field and examples of Zp. So if you could,please,simplify it a bit for simple proofs with explanations because i've just started it this week and it is knew for me.

Thank you.

4. 1)
fields are closed under two operations usually denoted $\displaystyle (F,+,\cdot)$. For $\displaystyle n\in \mathbb{Z}, a\in F$ $\displaystyle n\cdot a$just tells you to add a to itself n times just like multiplication in normal integers. $\displaystyle n\cdot a \in F$ because it is closed under addition, it has nothing to do with actually multiplying by n, it is just a notational way of saying add a n times.

2)fields must have a multiplicative identity (any nonzero element has a multiplicative inverse, this is what you get when you multiply an element by its inverse) this is what you call 1, (0 is reserved for the additive identity). So basically all you are doing is looking at a cyclic group generated by 1 under addition in F. $\displaystyle \{n\cdot 1|n\in \mathbb{Z} \}$ is this subgroup, furthermore if the characteristic of the field is p, then that means $\displaystyle p\cdot1=0$, so this group $\displaystyle \{n\cdot 1|n\in \mathbb{Z} \}$ is precisely $\displaystyle \mathbb{Z}_p$