The polynomial ring

Question 1

ring

Answer 1

abelian group under addition
associative under multiplication with identity
satisfies distributive laws

Question 2

commutative

Answer 2

ab=ba

Question 3

subring

Answer 3

1,-a,a+b,ab

Question 4

ideal

Answer 4

-a,a+b,ra

Question 5

principal ideal

Answer 5

I=Ra

Question 6

integral domain

Answer 6

R is non-empty and if ab=0 then a=0 or b=0

Question 7

principal ideal domain

Answer 7

integral domain and every ideal is principal

Question 8

ring homomorphism

Answer 8

φ(a+b)=φ(a)+φ(b)
φ(ab)=φ(a)φ(b)
φ(1)=1

Question 9

ring isomorphism

Answer 9

bijective ring homomorphism

Question 10

field

Answer 10

commutative ring where every non-zero a has an inverse

Question 11

subfield

Answer 11

subring and closed under inverses of non-zero elements
1,-a,a+b,ab,a^{-1}

Question 12

prime subfield

Answer 12

intersection of all subfields
(unique smallest subfield)

Question 13

prime field

Answer 13

no proper subfields

Question 14

degree

Answer 14

largest non-negative m st a_m is not zero

Question 15

f divides g

Answer 15

there exists h st g=fh

Question 16

group of units

Answer 16

collection of all elements with a multiplicative inverse

Question 17

irreducible

Answer 17

(non-trivial,non-unit) if r=st then s or t is a unit

Question 18

prime

Answer 18

if r|st then r|s or r|t

Question 19

root of polynomial

Answer 19

f(t)=0

Question 20

Gauss lemma

Answer 20

f,g in ZZ[t]
if p|fg then p|f or p|g

Question 21

Eisenstein irreducibility criterion

Answer 21

suppose there is p st
p does not divide an
p divides a0,…,an-1
p^2 does not divide a0
then f irreducible over Q