show that every field is euclidean domain

[david]

so this field must satisfies all conditions of euclidean domain

 

[stapel]

show that every field is euclidean domain, so this field must satisfies all conditions of euclidean domain

Where are you stuck in applying the definition?

 

[HallsofIvy]

The first thing you should do is write out those conditions. What are they?

 

[david]

there is a mapping v such that for a and b (b ≠ 0) in the Euclidean domain D
then there is a q and r such that
a = bq + r where r = 0 or v(r) < v(b) .


For any field, define
v(x) = 1 , for nonzero x in the field .
however, i just can't see how this will lead to a = bq + r where r=0 or v(r)<v(b)

 

[daon2]

there is a mapping v such that for a and b (b ≠ 0) in the Euclidean domain D
then there is a q and r such that
a = bq + r where r = 0 or v(r) < v(b) .


For any field, define
v(x) = 1 , for nonzero x in the field .
however, i just can't see how this will lead to a = bq + r where r=0 or v(r)<v(b)

What is the remainder when 1/2 is divided by 7/3?