Abstract We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in a standard form). This completes “minimal” arithmetic in free fields since “minimal” constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.

中文翻译：

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（某些）自由场中的标准形式：如何构造最小线性表示

摘要 我们描述了自由结合代数（在交换域上）的分数的普遍域中元素的标准形式。它是 Cohn 和 Reutenauer 提供的范式的特殊版本，可以使用线性代数技术为两个元素的和和乘积（以标准形式给出）构建最小线性表示（以标准形式） . 这完成了自由场中的“最小”算术，因为逆的“最小”构造是已知的。应用广泛：线性代数（在自由场上）、有理恒等式、计算两个非交换多项式的左 gcd 等。