grandes-ecoles 2014 QV.A

grandes-ecoles · France · centrale-maths2__pc Matrices Matrix Algebra and Product Properties

Let $(A, +, \times)$ be a commutative ring. For $p \in \mathbb{N}^*$, we denote by $C_p(A)$ the set of sums of $p$ squares of elements of $A$. Prove that for every ring $A$, the sets $C_p(A)$ are stable under multiplication when $p$ equals $1, 2, 4$ or $8$.
You may use the bilinear forms $B_p$ defined in part III and, if necessary, restrict yourself to the case where the ring $A$ is the ring $\mathbb{Z}$ of integers.

Let $(A, +, \times)$ be a commutative ring. For $p \in \mathbb{N}^*$, we denote by $C_p(A)$ the set of sums of $p$ squares of elements of $A$.\\
Prove that for every ring $A$, the sets $C_p(A)$ are stable under multiplication when $p$ equals $1, 2, 4$ or $8$.

You may use the bilinear forms $B_p$ defined in part III and, if necessary, restrict yourself to the case where the ring $A$ is the ring $\mathbb{Z}$ of integers.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.C.1 QI.C.2 QI.C.3 QI.D.1 QI.D.2 QI.E.1 QI.E.2 QI.E.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QV.A QV.B.1 QV.B.2 QV.B.3 QV.B.4