How to show that the two rational functions a ( t ) , b ( t ) &#x2208

arridsd9

arridsd9

Answered question

2022-06-15

How to show that the two rational functions a ( t ) , b ( t ) K ( t ) when Char ( K ) 2 are algebraically dependent.
(there is polynomial P ( x , y ) K [ x , y ] such that P ( a ( t ) , b ( t ) ) = 0)?

Answer & Explanation

benedictazk

benedictazk

Beginner2022-06-16Added 22 answers

Let us assume that a is not a constant. We may assume a ( t ) = c ( t ) d ( t ) , where c ( t ) , d ( t ) K [ t ] and d ( t ) 0. Now f ( x ) = c ( x ) d ( x ) a ( t ) K ( a ( t ) ) [ x ] which is not zero (why?), and f ( t ) = 0, this shows that t is algebraic over K ( a ). Therefore K ( t ) is an algebraic extension of K ( a ), in particular, b is algebraic over K ( a ).

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