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before second test

additive identity a+o=a
multiplicative identity ax=1
additive inverse a+x=o
multiplicative inverse [a][n]=1
subring ring closed under multiplication and subtraction
commutative ring xy=yx
unity multiplicative identity
units ab=ba=1 (a is a unit)
zero divisor R is a commutative ring, a does not equal 0, b does not equal 0, ab=0
integral domain commutative ring but has no zero divisors
field commutative ring with unity in which every non-zero element is a unit
factor of a polynomial something you can factor out with remainder 0
root of a polynomial plug in that number and get 0
Theorem 8.1 Suppose R is an integral domain and a,b,c are elements of R with a not equal to 0. If ab=ac, then b=c
Theorem 8.2 A field has no zero divisors
Theorem 8.5 Zn is a field iff n is prime
Theorem 8.6 Let o less than x less than m. Then [x] is a unit in the ring Zm iff gcd (x,m)=1
Theorem 8.8 All finite domains are fields
examples of finite fields Z5, Z3, Z2
examples of infinite field Q
finite integral domain that's not a field impossible
infinite integral domain that's not a field Z
a set that is closed under multiplication but not subtraction N
field without unity impossible
ring without unity 2Z (even integers)
polynomial in Q[x], reducible in Q[x] but has no roots in Q x^4 -4
Created by: monkeyhaha3