Field extensions

Question 1

field extension

Answer 1

pair of two fields where K is subfield of F

Question 2

tower law

Answer 2

ascending chain of fields
x basis for M over K, y basis for F over M. Then xy basis for F over K

Question 3

group

Answer 3

associativity, identity, inverse

Question 4

subgroup

Answer 4

1, ab, a^-1

Question 5

Galois group

Answer 5

group Aut_KK FF

Question 6

algebraic element over KK

Answer 6

there is non-zero polynomial in KK[x] such that f(a)=0

Question 7

transcendental

Answer 7

not algebraic

Question 8

minimal polynomial

Answer 8

monic polynomial of smallest degree such that m_a(a)=0

Question 9

simple extension

Answer 9

FF=KK(a)

Question 10

finitely generated

Answer 10

KK(a_1,…,a_n)

Question 11

finite

Answer 11

[FF:KK] finite

Question 12

algebraic extension

Answer 12

every element in FF is algebraic over KK

Question 13

subfield of algebraic elements in FF over KK

Answer 13

AA_FF/KK

Question 14

splitting field of f over KK

Answer 14

f=c(x-t1)…(x-tn) c in KK t1,…,tn in FF
FF=KK(t1,…,tn)

Question 15

uniqueness of the splitting field

Answer 15

isomorphic KKs, FFs splitting fields, then isomorphic