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F[x]/(x^2 + 1) is isomorphic to the field of complex numbers

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Modern Algebra
Ring Theory (XXXI)
Polynomial Ring
Irreducible Polynomial

Let F be a field of real numbers. Prove that F[x]/(x^2 + 1) is a field isomorphic to the field of complex numbers.

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It is the explanation of the following problem:

Let F be a field of real numbers. Then F[x]/(x^2 + 1) is a field isomorphic to the field of complex numbers.

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Modern Algebra
Ring Theory (XXXI)
Polynomial Ring
Irreducible Polynomial

By:- Thokchom Sarojkumar Sinha

Let be a field of real numbers. Prove that is a field isomorphic to the field of complex numbers.

Solution:- Let be an ideal in generated by .

is irreducible over the field of real numbers.

Then the ideal in is a maximal ideal.
Therefore is a field.
For if is a Euclidean ring (or a commutative ring with unit ...

Solution provided by:
Thokchom Sarojkumar Sinha, MSc
Education
  • BSc, Manipur University
  • MSc, Kanpur University
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