Hurkyl said:Once you've proven it for a field, it's easy to prove it for an integral domain.
If you tried to prove it directly for integral domains, it would be trickier -- the field case has fewer details to worry about.
A polynomial with coefficients in a field is a mathematical expression made up of variables, constants, and coefficients from a specific field. The field provides the set of numbers from which the coefficients can be chosen, and the variables represent unknown quantities. Examples of fields include the real numbers, complex numbers, and rational numbers.
A regular polynomial can have coefficients from any set of numbers, while a polynomial with coefficients in a field only uses coefficients from a specific field. This restriction allows for a more precise and generalized understanding of polynomials, as well as applications in various mathematical fields such as abstract algebra and number theory.
Some common fields used in polynomials include the real numbers, complex numbers, rational numbers, and finite fields such as the integers modulo a prime number. Other fields used in polynomials include algebraic number fields, function fields, and polynomial rings.
Polynomials with coefficients in a field have various applications in mathematics, including algebraic geometry, number theory, and signal processing. They are also used in constructing finite fields, factoring algorithms, and solving systems of equations. Additionally, they play a key role in computer science and coding theory.
Yes, polynomials with coefficients in a field can be graphed just like regular polynomials. The only difference is that the coefficients are chosen from a specific field instead of any set of numbers. This means that the graph may have different properties, such as being defined over a finite set of points instead of a continuous line. However, the fundamental concepts of graphing, such as finding roots and intercepts, still apply.