This article deals with the determination of the Euclidean minimum M (K) of a totally real number field K of degree n ≥ 2, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree 2 to 8 and small discriminants, most of which were previously unknown. Tables are given at the end of this paper. Recall that L(E K ) is a lattice of the hyperplane of R n defined by the equation 1≤i≤n x i = 0, which admits (L(ε i

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... ) 1≤i≤n−1 as a Z-basis, and that the kernel of L is {±1}. 2.2. Inhomogeneous minimum of a lattice. The notion of Euclidean minimum of a number field being closely related to the more geometrical notion of inhomogeneous minimum of a lattice, we recall some definitions and properties about the latter. Definition 2.2. Let R be a lattice of R n and let x be an element of R n . The real number is called the inhomogeneous minimum of x (for the product form) relative to R. Proposition 2.1. m R has the following properties: for all x ∈ R n and all X ∈ R. iii) m R is upper semi-continuous on R n , and on R n /R. Proof. For i) and iii) see [Ca]. For ii), let f denote the embedding of Moreover, from the definition of m R and z.R = R we have m R (z·x) = N (z)m R (x), and i) gives the result. Property iii) of Proposition 2.1 has interesting consequences. Corollary 2.3. m R is bounded and attains its maximum at an x ∈ R n (at least one modulo R). Now we need some more definitions. Definition 2.3. The inhomogeneous minimum of R, denoted by m(R), is the real number defined by Definition 2.4. If x ∈ R n is such that m R (x) = m(R), we shall say that x is critical. Euclidean minimum of K. Let us now recall some basic facts about the Euclidean minimum of the number field K. Definition 2.5. Let ξ ∈ K. The Euclidean minimum of ξ (relative to the norm) is the real number M K (ξ) defined by M K (ξ) = inf |N K/Q (ξ − Υ)|; Υ ∈ Z K . M K has the following properties.