grandes-ecoles 2019 Q19

grandes-ecoles · France · x-ens-maths1__mp Number Theory Algebraic Number Theory and Minimal Polynomials

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Verify that $P _ { n }$ has no root in $\mathbb { Q }$ and that $P _ { n }$ has at least one real root strictly greater than 1. We fix such a root $\alpha _ { n }$ in the sequel.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by
$$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$
Verify that $P _ { n }$ has no root in $\mathbb { Q }$ and that $P _ { n }$ has at least one real root strictly greater than 1. We fix such a root $\alpha _ { n }$ in the sequel.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24