Answered: If a and b belong to Z[√d], where d is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

If a and b belong to Z[√d], where d is not divisible by the square of
a prime and ab is a unit, prove that a and b are units.

Expert Solution

Step 1

Let $d$ be an integer that is not divisible by the square of a prime.

Let $a$ and $b$ belong to $Z [\sqrt{d}]$ and $a b$ is a unit.

Here we need to that $a$ and $b$ are units.

Now, claim the statement,

"The only units of $Z [\sqrt{d}]$ are $+ 1$ and $- 1$ ".

It is given that $N (a + b \sqrt{d}) = |a^{2} - b d^{2}|$ .

Let $x = a + b \sqrt{d} \in Z [\sqrt{d}]$ be a unit in $Z [\sqrt{d}]$ .

Note that $x$ is a unit in $Z [\sqrt{d}]$ if and only if $N (x) = 1$ .

So,

$\begin{array}{rcl} N (x) & = & N (a + b \sqrt{d}) \\ = & |a^{2} - d b^{2}| \\ = & 1 \end{array}$

Step by step

Solved in 2 steps

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

If a and b belong to Z[√d], where d is not divisible by the square ofa prime and ab is a unit, prove that a and b are units.