#### Vol. 7, No. 7, 2013

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Cohomological invariants of algebraic tori

### Sam Blinstein and Alexander Merkurjev

Vol. 7 (2013), No. 7, 1643–1684
##### Abstract

Let $G$ be an algebraic group over a field $F$. As defined by Serre, a cohomological invariant of $G$ of degree $n$ with values in $ℚ∕ℤ\left(j\right)$ is a functorial-in-$K$ collection of maps of sets ${Tors}_{G}\left(K\right)\to {H}^{n}\left(K,ℚ∕ℤ\left(j\right)\right)$ for all field extensions $K∕F$, where ${Tors}_{G}\left(K\right)$ is the set of isomorphism classes of $G$-torsors over Spec $K$. We study the group of degree $3$ invariants of an algebraic torus with values in $ℚ∕ℤ\left(2\right)$. In particular, we compute the group ${H}_{nr}^{3}\left(F\left(S\right),ℚ∕ℤ\left(2\right)\right)$ of unramified cohomology of an algebraic torus $S$.

##### Keywords
algebraic tori, cohomological invariants, Galois cohomology
Primary: 11E72
Secondary: 12G05