| 1. | A field with no nontrivial algebraic extensions is called algebraically closed.
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| 2. | Obtained by extending the identity map of the algebraic tensor product.
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| 3. | The second column gives an equivalent algebraic formulation of the problem.
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| 4. | This led to modern abstract algebraic notions such as Euclidean domains.
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| 5. | This result shows that the finiteness restriction can have algebraic consequences.
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| 6. | Just as the automorphisms of an algebraic structure form a heap.
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| 7. | Further branches crucially applying groups include algebraic geometry and number theory.
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| 8. | This reduces the problem to the solution of an algebraic equation.
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| 9. | A similar definition can be made for certain other algebraic structures.
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| 10. | In the following discussion, a simplified algebraic model is used.
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