An algebraic closure of a field is a larger field in which every polynomial equation with coefficients from the original field has a solution. For example, the algebraic closure of the field of real numbers includes all complex numbers, allowing for solutions to equations like x^2 + 1 = 0. In essence, an algebraic closure ensures that any polynomial can be factored completely into linear factors. This property is crucial in abstract algebra and field theory, as it allows mathematicians to study the roots of polynomials in a comprehensive way, facilitating deeper insights into their structure and behavior.